diff --git a/asv_bench/benchmarks/strftime.py b/asv_bench/benchmarks/strftime.py
index 47f25b331ab9b..5f2832e361eb5 100644
--- a/asv_bench/benchmarks/strftime.py
+++ b/asv_bench/benchmarks/strftime.py
@@ -79,12 +79,6 @@ def time_frame_period_formatting_default(self, nobs, freq):
     def time_frame_period_formatting_default_explicit(self, nobs, freq):
         self.data["p"].dt.strftime(date_format=self.default_fmt)
 
-    def time_frame_period_formatting_index_default(self, nobs, freq):
-        self.data.index.format()
-
-    def time_frame_period_formatting_index_default_explicit(self, nobs, freq):
-        self.data.index.format(self.default_fmt)
-
     def time_frame_period_formatting_custom(self, nobs, freq):
         self.data["p"].dt.strftime(date_format="%Y-%m-%d --- %H:%M:%S")
 
diff --git a/doc/redirects.csv b/doc/redirects.csv
index 1f12ba08b8005..5d351680ec03a 100644
--- a/doc/redirects.csv
+++ b/doc/redirects.csv
@@ -118,7 +118,6 @@ generated/pandas.api.types.is_int64_dtype,../reference/api/pandas.api.types.is_i
 generated/pandas.api.types.is_integer_dtype,../reference/api/pandas.api.types.is_integer_dtype
 generated/pandas.api.types.is_integer,../reference/api/pandas.api.types.is_integer
 generated/pandas.api.types.is_interval_dtype,../reference/api/pandas.api.types.is_interval_dtype
-generated/pandas.api.types.is_interval,../reference/api/pandas.api.types.is_interval
 generated/pandas.api.types.is_iterator,../reference/api/pandas.api.types.is_iterator
 generated/pandas.api.types.is_list_like,../reference/api/pandas.api.types.is_list_like
 generated/pandas.api.types.is_named_tuple,../reference/api/pandas.api.types.is_named_tuple
diff --git a/doc/source/reference/arrays.rst b/doc/source/reference/arrays.rst
index 93b56d1b48c02..a631cd517e3c2 100644
--- a/doc/source/reference/arrays.rst
+++ b/doc/source/reference/arrays.rst
@@ -700,7 +700,6 @@ Scalar introspection
     api.types.is_float
     api.types.is_hashable
     api.types.is_integer
-    api.types.is_interval
     api.types.is_number
     api.types.is_re
     api.types.is_re_compilable
diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index 3bfe16b29535d..9423adc7b148d 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -108,12 +108,15 @@ Removal of prior version deprecations/changes
 - Removed ``DataFrame.first`` and ``DataFrame.last`` (:issue:`53710`)
 - Removed ``DataFrameGroupBy.grouper`` and ``SeriesGroupBy.grouper`` (:issue:`56521`)
 - Removed ``DataFrameGroupby.fillna`` and ``SeriesGroupBy.fillna``` (:issue:`55719`)
+- Removed ``Index.format``, use :meth:`Index.astype` with ``str`` or :meth:`Index.map` with a ``formatter`` function instead (:issue:`55439`)
 - Removed ``Series.__int__`` and ``Series.__float__``. Call ``int(Series.iloc[0])`` or ``float(Series.iloc[0])`` instead. (:issue:`51131`)
 - Removed ``Series.ravel`` (:issue:`56053`)
 - Removed ``Series.view`` (:issue:`56054`)
 - Removed ``axis`` argument from :meth:`DataFrame.groupby`, :meth:`Series.groupby`, :meth:`DataFrame.rolling`, :meth:`Series.rolling`, :meth:`DataFrame.resample`, and :meth:`Series.resample` (:issue:`51203`)
 - Removed ``axis`` argument from all groupby operations (:issue:`50405`)
+- Removed ``pandas.api.types.is_interval`` and ``pandas.api.types.is_period``, use ``isinstance(obj, pd.Interval)`` and ``isinstance(obj, pd.Period)`` instead (:issue:`55264`)
 - Removed ``pandas.io.sql.execute`` (:issue:`50185`)
+- Removed ``pandas.value_counts``, use :meth:`Series.value_counts` instead (:issue:`53493`)
 - Removed ``read_gbq`` and ``DataFrame.to_gbq``. Use ``pandas_gbq.read_gbq`` and ``pandas_gbq.to_gbq`` instead https://pandas-gbq.readthedocs.io/en/latest/api.html (:issue:`55525`)
 - Removed deprecated argument ``obj`` in :meth:`.DataFrameGroupBy.get_group` and :meth:`.SeriesGroupBy.get_group` (:issue:`53545`)
 - Removed the ``ArrayManager`` (:issue:`55043`)
diff --git a/pandas/__init__.py b/pandas/__init__.py
index 95601f226a83f..7b5b00d92db1f 100644
--- a/pandas/__init__.py
+++ b/pandas/__init__.py
@@ -101,7 +101,6 @@
     Grouper,
     factorize,
     unique,
-    value_counts,
     NamedAgg,
     array,
     Categorical,
@@ -378,6 +377,5 @@
     "to_timedelta",
     "tseries",
     "unique",
-    "value_counts",
     "wide_to_long",
 ]
diff --git a/pandas/_libs/lib.pyi b/pandas/_libs/lib.pyi
index c8befabbf86de..34193a9b1d231 100644
--- a/pandas/_libs/lib.pyi
+++ b/pandas/_libs/lib.pyi
@@ -14,8 +14,6 @@ from typing import (
 
 import numpy as np
 
-from pandas._libs.interval import Interval
-from pandas._libs.tslibs import Period
 from pandas._typing import (
     ArrayLike,
     DtypeObj,
@@ -44,8 +42,6 @@ def is_iterator(obj: object) -> bool: ...
 def is_scalar(val: object) -> bool: ...
 def is_list_like(obj: object, allow_sets: bool = ...) -> bool: ...
 def is_pyarrow_array(obj: object) -> bool: ...
-def is_period(val: object) -> TypeGuard[Period]: ...
-def is_interval(obj: object) -> TypeGuard[Interval]: ...
 def is_decimal(obj: object) -> TypeGuard[Decimal]: ...
 def is_complex(obj: object) -> TypeGuard[complex]: ...
 def is_bool(obj: object) -> TypeGuard[bool | np.bool_]: ...
diff --git a/pandas/_libs/lib.pyx b/pandas/_libs/lib.pyx
index 50feda8fb188e..3146b31a2e7e8 100644
--- a/pandas/_libs/lib.pyx
+++ b/pandas/_libs/lib.pyx
@@ -1164,43 +1164,6 @@ cpdef bint is_decimal(object obj):
     return isinstance(obj, Decimal)
 
 
-cpdef bint is_interval(object obj):
-    import warnings
-
-    from pandas.util._exceptions import find_stack_level
-
-    warnings.warn(
-        # GH#55264
-        "is_interval is deprecated and will be removed in a future version. "
-        "Use isinstance(obj, pd.Interval) instead.",
-        FutureWarning,
-        stacklevel=find_stack_level(),
-    )
-    return getattr(obj, "_typ", "_typ") == "interval"
-
-
-def is_period(val: object) -> bool:
-    """
-    Return True if given object is Period.
-
-    Returns
-    -------
-    bool
-    """
-    import warnings
-
-    from pandas.util._exceptions import find_stack_level
-
-    warnings.warn(
-        # GH#55264
-        "is_period is deprecated and will be removed in a future version. "
-        "Use isinstance(obj, pd.Period) instead.",
-        FutureWarning,
-        stacklevel=find_stack_level(),
-    )
-    return is_period_object(val)
-
-
 def is_list_like(obj: object, allow_sets: bool = True) -> bool:
     """
     Check if the object is list-like.
diff --git a/pandas/core/algorithms.py b/pandas/core/algorithms.py
index b346cb9b2c175..500f805499a8b 100644
--- a/pandas/core/algorithms.py
+++ b/pandas/core/algorithms.py
@@ -818,53 +818,6 @@ def factorize(
     return codes, uniques
 
 
-def value_counts(
-    values,
-    sort: bool = True,
-    ascending: bool = False,
-    normalize: bool = False,
-    bins=None,
-    dropna: bool = True,
-) -> Series:
-    """
-    Compute a histogram of the counts of non-null values.
-
-    Parameters
-    ----------
-    values : ndarray (1-d)
-    sort : bool, default True
-        Sort by values
-    ascending : bool, default False
-        Sort in ascending order
-    normalize: bool, default False
-        If True then compute a relative histogram
-    bins : integer, optional
-        Rather than count values, group them into half-open bins,
-        convenience for pd.cut, only works with numeric data
-    dropna : bool, default True
-        Don't include counts of NaN
-
-    Returns
-    -------
-    Series
-    """
-    warnings.warn(
-        # GH#53493
-        "pandas.value_counts is deprecated and will be removed in a "
-        "future version. Use pd.Series(obj).value_counts() instead.",
-        FutureWarning,
-        stacklevel=find_stack_level(),
-    )
-    return value_counts_internal(
-        values,
-        sort=sort,
-        ascending=ascending,
-        normalize=normalize,
-        bins=bins,
-        dropna=dropna,
-    )
-
-
 def value_counts_internal(
     values,
     sort: bool = True,
diff --git a/pandas/core/api.py b/pandas/core/api.py
index 2cfe5ffc0170d..3d2e855831c05 100644
--- a/pandas/core/api.py
+++ b/pandas/core/api.py
@@ -23,7 +23,6 @@
 from pandas.core.algorithms import (
     factorize,
     unique,
-    value_counts,
 )
 from pandas.core.arrays import Categorical
 from pandas.core.arrays.boolean import BooleanDtype
@@ -136,5 +135,4 @@
     "UInt64Dtype",
     "UInt8Dtype",
     "unique",
-    "value_counts",
 ]
diff --git a/pandas/core/dtypes/api.py b/pandas/core/dtypes/api.py
index 254abe330b8e7..e66104d6afcd9 100644
--- a/pandas/core/dtypes/api.py
+++ b/pandas/core/dtypes/api.py
@@ -20,7 +20,6 @@
     is_int64_dtype,
     is_integer,
     is_integer_dtype,
-    is_interval,
     is_interval_dtype,
     is_iterator,
     is_list_like,
@@ -63,7 +62,6 @@
     "is_int64_dtype",
     "is_integer",
     "is_integer_dtype",
-    "is_interval",
     "is_interval_dtype",
     "is_iterator",
     "is_list_like",
diff --git a/pandas/core/dtypes/common.py b/pandas/core/dtypes/common.py
index 99114d996cc4c..8eee15e13791a 100644
--- a/pandas/core/dtypes/common.py
+++ b/pandas/core/dtypes/common.py
@@ -42,7 +42,6 @@
     is_float,
     is_hashable,
     is_integer,
-    is_interval,
     is_iterator,
     is_list_like,
     is_named_tuple,
@@ -1710,7 +1709,6 @@ def is_all_strings(value: ArrayLike) -> bool:
     "is_float_dtype",
     "is_int64_dtype",
     "is_integer_dtype",
-    "is_interval",
     "is_interval_dtype",
     "is_iterator",
     "is_named_tuple",
diff --git a/pandas/core/dtypes/inference.py b/pandas/core/dtypes/inference.py
index c0d9b418b9e79..bd4e5182b24bf 100644
--- a/pandas/core/dtypes/inference.py
+++ b/pandas/core/dtypes/inference.py
@@ -29,8 +29,6 @@
 
 is_decimal = lib.is_decimal
 
-is_interval = lib.is_interval
-
 is_list_like = lib.is_list_like
 
 is_iterator = lib.is_iterator
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 83290f7900ab6..68fd49bd6fb54 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -1394,36 +1394,6 @@ def _mpl_repr(self) -> np.ndarray:
             return cast(np.ndarray, self.values)
         return self.astype(object, copy=False)._values
 
-    def format(
-        self,
-        name: bool = False,
-        formatter: Callable | None = None,
-        na_rep: str_t = "NaN",
-    ) -> list[str_t]:
-        """
-        Render a string representation of the Index.
-        """
-        warnings.warn(
-            # GH#55413
-            f"{type(self).__name__}.format is deprecated and will be removed "
-            "in a future version. Convert using index.astype(str) or "
-            "index.map(formatter) instead.",
-            FutureWarning,
-            stacklevel=find_stack_level(),
-        )
-        header = []
-        if name:
-            header.append(
-                pprint_thing(self.name, escape_chars=("\t", "\r", "\n"))
-                if self.name is not None
-                else ""
-            )
-
-        if formatter is not None:
-            return header + list(self.map(formatter))
-
-        return self._format_with_header(header=header, na_rep=na_rep)
-
     _default_na_rep = "NaN"
 
     @final
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 451adce4b320e..117de2e6f950c 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -10,11 +10,9 @@
 from typing import (
     TYPE_CHECKING,
     Any,
-    Callable,
     cast,
     final,
 )
-import warnings
 
 import numpy as np
 
@@ -43,7 +41,6 @@
     cache_readonly,
     doc,
 )
-from pandas.util._exceptions import find_stack_level
 
 from pandas.core.dtypes.common import (
     is_integer,
@@ -191,39 +188,6 @@ def _convert_tolerance(self, tolerance, target):
     # Rendering Methods
     _default_na_rep = "NaT"
 
-    def format(
-        self,
-        name: bool = False,
-        formatter: Callable | None = None,
-        na_rep: str = "NaT",
-        date_format: str | None = None,
-    ) -> list[str]:
-        """
-        Render a string representation of the Index.
-        """
-        warnings.warn(
-            # GH#55413
-            f"{type(self).__name__}.format is deprecated and will be removed "
-            "in a future version. Convert using index.astype(str) or "
-            "index.map(formatter) instead.",
-            FutureWarning,
-            stacklevel=find_stack_level(),
-        )
-        header = []
-        if name:
-            header.append(
-                ibase.pprint_thing(self.name, escape_chars=("\t", "\r", "\n"))
-                if self.name is not None
-                else ""
-            )
-
-        if formatter is not None:
-            return header + list(self.map(formatter))
-
-        return self._format_with_header(
-            header=header, na_rep=na_rep, date_format=date_format
-        )
-
     def _format_with_header(
         self, *, header: list[str], na_rep: str, date_format: str | None = None
     ) -> list[str]:
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index c81d76d471a5f..9c6156b97689e 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -107,10 +107,7 @@
     lexsort_indexer,
 )
 
-from pandas.io.formats.printing import (
-    get_adjustment,
-    pprint_thing,
-)
+from pandas.io.formats.printing import pprint_thing
 
 if TYPE_CHECKING:
     from pandas import (
@@ -1421,90 +1418,6 @@ def _get_values_for_csv(
             )
             return mi._values
 
-    def format(
-        self,
-        name: bool | None = None,
-        formatter: Callable | None = None,
-        na_rep: str | None = None,
-        names: bool = False,
-        space: int = 2,
-        sparsify=None,
-        adjoin: bool = True,
-    ) -> list:
-        warnings.warn(
-            # GH#55413
-            f"{type(self).__name__}.format is deprecated and will be removed "
-            "in a future version. Convert using index.astype(str) or "
-            "index.map(formatter) instead.",
-            FutureWarning,
-            stacklevel=find_stack_level(),
-        )
-
-        if name is not None:
-            names = name
-
-        if len(self) == 0:
-            return []
-
-        formatted: Iterable
-        stringified_levels = []
-        for lev, level_codes in zip(self.levels, self.codes):
-            na = na_rep if na_rep is not None else _get_na_rep(lev.dtype)
-
-            if len(lev) > 0:
-                formatted = lev.take(level_codes).format(formatter=formatter)
-
-                # we have some NA
-                mask = level_codes == -1
-                if mask.any():
-                    formatted = np.array(formatted, dtype=object)
-                    formatted[mask] = na
-                    formatted = formatted.tolist()
-
-            else:
-                # weird all NA case
-                formatted = [
-                    pprint_thing(na if isna(x) else x, escape_chars=("\t", "\r", "\n"))
-                    for x in algos.take_nd(lev._values, level_codes)
-                ]
-            stringified_levels.append(formatted)
-
-        result_levels = []
-        # Incompatible types in assignment (expression has type "Iterable[Any]",
-        # variable has type "Index")
-        for lev, lev_name in zip(stringified_levels, self.names):  # type: ignore[assignment]
-            level = []
-
-            if names:
-                level.append(
-                    pprint_thing(lev_name, escape_chars=("\t", "\r", "\n"))
-                    if lev_name is not None
-                    else ""
-                )
-
-            level.extend(np.array(lev, dtype=object))
-            result_levels.append(level)
-
-        if sparsify is None:
-            sparsify = get_option("display.multi_sparse")
-
-        if sparsify:
-            sentinel: Literal[""] | bool | lib.NoDefault = ""
-            # GH3547 use value of sparsify as sentinel if it's "Falsey"
-            assert isinstance(sparsify, bool) or sparsify is lib.no_default
-            if sparsify in [False, lib.no_default]:
-                sentinel = sparsify
-            # little bit of a kludge job for #1217
-            result_levels = sparsify_labels(
-                result_levels, start=int(names), sentinel=sentinel
-            )
-
-        if adjoin:
-            adj = get_adjustment()
-            return adj.adjoin(space, *result_levels).split("\n")
-        else:
-            return result_levels
-
     def _format_multi(
         self,
         *,
diff --git a/pandas/tests/api/test_api.py b/pandas/tests/api/test_api.py
index 6ed1d07d0cc3d..15b6c9abaea8f 100644
--- a/pandas/tests/api/test_api.py
+++ b/pandas/tests/api/test_api.py
@@ -133,7 +133,6 @@ class TestPDApi(Base):
         "show_versions",
         "timedelta_range",
         "unique",
-        "value_counts",
         "wide_to_long",
     ]
 
@@ -292,7 +291,6 @@ class TestApi(Base):
         "is_int64_dtype",
         "is_integer",
         "is_integer_dtype",
-        "is_interval",
         "is_interval_dtype",
         "is_iterator",
         "is_list_like",
diff --git a/pandas/tests/api/test_types.py b/pandas/tests/api/test_types.py
index fbaa6e7e18bca..bf39370c49d76 100644
--- a/pandas/tests/api/test_types.py
+++ b/pandas/tests/api/test_types.py
@@ -34,7 +34,6 @@ class TestTypes(Base):
         "is_timedelta64_ns_dtype",
         "is_unsigned_integer_dtype",
         "is_period_dtype",
-        "is_interval",
         "is_interval_dtype",
         "is_re",
         "is_re_compilable",
diff --git a/pandas/tests/dtypes/test_inference.py b/pandas/tests/dtypes/test_inference.py
index f1f5cb1620345..0434ad7e50568 100644
--- a/pandas/tests/dtypes/test_inference.py
+++ b/pandas/tests/dtypes/test_inference.py
@@ -63,7 +63,6 @@
     Index,
     Interval,
     Period,
-    PeriodIndex,
     Series,
     Timedelta,
     TimedeltaIndex,
@@ -1617,25 +1616,6 @@ def test_to_object_array_width(self):
         out = lib.to_object_array(rows, min_width=5)
         tm.assert_numpy_array_equal(out, expected)
 
-    def test_is_period(self):
-        # GH#55264
-        msg = "is_period is deprecated and will be removed in a future version"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert lib.is_period(Period("2011-01", freq="M"))
-            assert not lib.is_period(PeriodIndex(["2011-01"], freq="M"))
-            assert not lib.is_period(Timestamp("2011-01"))
-            assert not lib.is_period(1)
-            assert not lib.is_period(np.nan)
-
-    def test_is_interval(self):
-        # GH#55264
-        msg = "is_interval is deprecated and will be removed in a future version"
-        item = Interval(1, 2)
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert lib.is_interval(item)
-            assert not lib.is_interval(pd.IntervalIndex([item]))
-            assert not lib.is_interval(pd.IntervalIndex([item])._engine)
-
     def test_categorical(self):
         # GH 8974
         arr = Categorical(list("abc"))
diff --git a/pandas/tests/indexes/base_class/test_formats.py b/pandas/tests/indexes/base_class/test_formats.py
index f30b578cfcf56..4580e00069dc1 100644
--- a/pandas/tests/indexes/base_class/test_formats.py
+++ b/pandas/tests/indexes/base_class/test_formats.py
@@ -144,20 +144,6 @@ def test_summary_bug(self):
     def test_index_repr_bool_nan(self):
         # GH32146
         arr = Index([True, False, np.nan], dtype=object)
-        msg = "Index.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            exp1 = arr.format()
-        out1 = ["True", "False", "NaN"]
-        assert out1 == exp1
-
         exp2 = repr(arr)
         out2 = "Index([True, False, nan], dtype='object')"
         assert out2 == exp2
-
-    def test_format_different_scalar_lengths(self):
-        # GH#35439
-        idx = Index(["aaaaaaaaa", "b"])
-        expected = ["aaaaaaaaa", "b"]
-        msg = r"Index\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert idx.format() == expected
diff --git a/pandas/tests/indexes/categorical/test_formats.py b/pandas/tests/indexes/categorical/test_formats.py
index 522ca1bc2afde..74e738a543300 100644
--- a/pandas/tests/indexes/categorical/test_formats.py
+++ b/pandas/tests/indexes/categorical/test_formats.py
@@ -7,18 +7,9 @@
 import pandas._config.config as cf
 
 from pandas import CategoricalIndex
-import pandas._testing as tm
 
 
 class TestCategoricalIndexRepr:
-    def test_format_different_scalar_lengths(self):
-        # GH#35439
-        idx = CategoricalIndex(["aaaaaaaaa", "b"])
-        expected = ["aaaaaaaaa", "b"]
-        msg = r"CategoricalIndex\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert idx.format() == expected
-
     @pytest.mark.xfail(using_pyarrow_string_dtype(), reason="repr different")
     def test_string_categorical_index_repr(self):
         # short
diff --git a/pandas/tests/indexes/datetimes/test_formats.py b/pandas/tests/indexes/datetimes/test_formats.py
index f8b01f7985535..6e4e22942ab07 100644
--- a/pandas/tests/indexes/datetimes/test_formats.py
+++ b/pandas/tests/indexes/datetimes/test_formats.py
@@ -286,66 +286,3 @@ def test_dti_business_repr_etc_smoke(self, tz, freq):
         repr(dti)
         dti._summary()
         dti[2:2]._summary()
-
-
-class TestFormat:
-    def test_format(self):
-        # GH#35439
-        idx = pd.date_range("20130101", periods=5)
-        expected = [f"{x:%Y-%m-%d}" for x in idx]
-        msg = r"DatetimeIndex\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert idx.format() == expected
-
-    def test_format_with_name_time_info(self):
-        # bug I fixed 12/20/2011
-        dates = pd.date_range("2011-01-01 04:00:00", periods=10, name="something")
-
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = dates.format(name=True)
-        assert formatted[0] == "something"
-
-    def test_format_datetime_with_time(self):
-        dti = DatetimeIndex([datetime(2012, 2, 7), datetime(2012, 2, 7, 23)])
-
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = dti.format()
-        expected = ["2012-02-07 00:00:00", "2012-02-07 23:00:00"]
-        assert len(result) == 2
-        assert result == expected
-
-    def test_format_datetime(self):
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = pd.to_datetime([datetime(2003, 1, 1, 12), NaT]).format()
-        assert formatted[0] == "2003-01-01 12:00:00"
-        assert formatted[1] == "NaT"
-
-    def test_format_date(self):
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = pd.to_datetime([datetime(2003, 1, 1), NaT]).format()
-        assert formatted[0] == "2003-01-01"
-        assert formatted[1] == "NaT"
-
-    def test_format_date_tz(self):
-        dti = pd.to_datetime([datetime(2013, 1, 1)], utc=True)
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = dti.format()
-        assert formatted[0] == "2013-01-01 00:00:00+00:00"
-
-        dti = pd.to_datetime([datetime(2013, 1, 1), NaT], utc=True)
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = dti.format()
-        assert formatted[0] == "2013-01-01 00:00:00+00:00"
-
-    def test_format_date_explicit_date_format(self):
-        dti = pd.to_datetime([datetime(2003, 2, 1), NaT])
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = dti.format(date_format="%m-%d-%Y", na_rep="UT")
-        assert formatted[0] == "02-01-2003"
-        assert formatted[1] == "UT"
diff --git a/pandas/tests/indexes/multi/test_formats.py b/pandas/tests/indexes/multi/test_formats.py
index 286a048f02dad..cc6a33c22503d 100644
--- a/pandas/tests/indexes/multi/test_formats.py
+++ b/pandas/tests/indexes/multi/test_formats.py
@@ -6,48 +6,6 @@
     Index,
     MultiIndex,
 )
-import pandas._testing as tm
-
-
-def test_format(idx):
-    msg = "MultiIndex.format is deprecated"
-    with tm.assert_produces_warning(FutureWarning, match=msg):
-        idx.format()
-        idx[:0].format()
-
-
-def test_format_integer_names():
-    index = MultiIndex(
-        levels=[[0, 1], [0, 1]], codes=[[0, 0, 1, 1], [0, 1, 0, 1]], names=[0, 1]
-    )
-    msg = "MultiIndex.format is deprecated"
-    with tm.assert_produces_warning(FutureWarning, match=msg):
-        index.format(names=True)
-
-
-def test_format_sparse_config(idx):
-    # GH1538
-    msg = "MultiIndex.format is deprecated"
-    with pd.option_context("display.multi_sparse", False):
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = idx.format()
-    assert result[1] == "foo  two"
-
-
-def test_format_sparse_display():
-    index = MultiIndex(
-        levels=[[0, 1], [0, 1], [0, 1], [0]],
-        codes=[
-            [0, 0, 0, 1, 1, 1],
-            [0, 0, 1, 0, 0, 1],
-            [0, 1, 0, 0, 1, 0],
-            [0, 0, 0, 0, 0, 0],
-        ],
-    )
-    msg = "MultiIndex.format is deprecated"
-    with tm.assert_produces_warning(FutureWarning, match=msg):
-        result = index.format()
-    assert result[3] == "1  0  0  0"
 
 
 def test_repr_with_unicode_data():
diff --git a/pandas/tests/indexes/period/test_formats.py b/pandas/tests/indexes/period/test_formats.py
index 81c79f7d18f2f..9f36eb1e7a1d1 100644
--- a/pandas/tests/indexes/period/test_formats.py
+++ b/pandas/tests/indexes/period/test_formats.py
@@ -1,10 +1,3 @@
-from contextlib import nullcontext
-from datetime import (
-    datetime,
-    time,
-)
-import locale
-
 import numpy as np
 import pytest
 
@@ -16,13 +9,6 @@
 import pandas._testing as tm
 
 
-def get_local_am_pm():
-    """Return the AM and PM strings returned by strftime in current locale."""
-    am_local = time(1).strftime("%p")
-    pm_local = time(13).strftime("%p")
-    return am_local, pm_local
-
-
 def test_get_values_for_csv():
     index = PeriodIndex(["2017-01-01", "2017-01-02", "2017-01-03"], freq="D")
 
@@ -56,15 +42,6 @@ def test_get_values_for_csv():
 
 
 class TestPeriodIndexRendering:
-    def test_format_empty(self):
-        # GH#35712
-        empty_idx = PeriodIndex([], freq="Y")
-        msg = r"PeriodIndex\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format() == []
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format(name=True) == [""]
-
     @pytest.mark.parametrize("method", ["__repr__", "__str__"])
     def test_representation(self, method):
         # GH#7601
@@ -215,136 +192,3 @@ def test_summary(self):
         ):
             result = idx._summary()
             assert result == expected
-
-
-class TestPeriodIndexFormat:
-    def test_period_format_and_strftime_default(self):
-        per = PeriodIndex([datetime(2003, 1, 1, 12), None], freq="h")
-
-        # Default formatting
-        msg = "PeriodIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = per.format()
-        assert formatted[0] == "2003-01-01 12:00"  # default: minutes not shown
-        assert formatted[1] == "NaT"
-        # format is equivalent to strftime(None)...
-        assert formatted[0] == per.strftime(None)[0]
-        assert per.strftime(None)[1] is np.nan  # ...except for NaTs
-
-        # Same test with nanoseconds freq
-        per = pd.period_range("2003-01-01 12:01:01.123456789", periods=2, freq="ns")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = per.format()
-        assert (formatted == per.strftime(None)).all()
-        assert formatted[0] == "2003-01-01 12:01:01.123456789"
-        assert formatted[1] == "2003-01-01 12:01:01.123456790"
-
-    def test_period_custom(self):
-        # GH#46252 custom formatting directives %l (ms) and %u (us)
-        msg = "PeriodIndex.format is deprecated"
-
-        # 3 digits
-        per = pd.period_range("2003-01-01 12:01:01.123", periods=2, freq="ms")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
-        assert formatted[0] == "03 12:01:01 (ms=123 us=123000 ns=123000000)"
-        assert formatted[1] == "03 12:01:01 (ms=124 us=124000 ns=124000000)"
-
-        # 6 digits
-        per = pd.period_range("2003-01-01 12:01:01.123456", periods=2, freq="us")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
-        assert formatted[0] == "03 12:01:01 (ms=123 us=123456 ns=123456000)"
-        assert formatted[1] == "03 12:01:01 (ms=123 us=123457 ns=123457000)"
-
-        # 9 digits
-        per = pd.period_range("2003-01-01 12:01:01.123456789", periods=2, freq="ns")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = per.format(date_format="%y %I:%M:%S (ms=%l us=%u ns=%n)")
-        assert formatted[0] == "03 12:01:01 (ms=123 us=123456 ns=123456789)"
-        assert formatted[1] == "03 12:01:01 (ms=123 us=123456 ns=123456790)"
-
-    def test_period_tz(self):
-        # Formatting periods created from a datetime with timezone.
-        msg = r"PeriodIndex\.format is deprecated"
-        # This timestamp is in 2013 in Europe/Paris but is 2012 in UTC
-        dt = pd.to_datetime(["2013-01-01 00:00:00+01:00"], utc=True)
-
-        # Converting to a period looses the timezone information
-        # Since tz is currently set as utc, we'll see 2012
-        with tm.assert_produces_warning(UserWarning, match="will drop timezone"):
-            per = dt.to_period(freq="h")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert per.format()[0] == "2012-12-31 23:00"
-
-        # If tz is currently set as paris before conversion, we'll see 2013
-        dt = dt.tz_convert("Europe/Paris")
-        with tm.assert_produces_warning(UserWarning, match="will drop timezone"):
-            per = dt.to_period(freq="h")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert per.format()[0] == "2013-01-01 00:00"
-
-    @pytest.mark.parametrize(
-        "locale_str",
-        [
-            pytest.param(None, id=str(locale.getlocale())),
-            "it_IT.utf8",
-            "it_IT",  # Note: encoding will be 'ISO8859-1'
-            "zh_CN.utf8",
-            "zh_CN",  # Note: encoding will be 'gb2312'
-        ],
-    )
-    def test_period_non_ascii_fmt(self, locale_str):
-        # GH#46468 non-ascii char in input format string leads to wrong output
-
-        # Skip if locale cannot be set
-        if locale_str is not None and not tm.can_set_locale(locale_str, locale.LC_ALL):
-            pytest.skip(f"Skipping as locale '{locale_str}' cannot be set on host.")
-
-        # Change locale temporarily for this test.
-        with tm.set_locale(locale_str, locale.LC_ALL) if locale_str else nullcontext():
-            # Scalar
-            per = pd.Period("2018-03-11 13:00", freq="h")
-            assert per.strftime("%y é") == "18 é"
-
-            # Index
-            per = pd.period_range("2003-01-01 01:00:00", periods=2, freq="12h")
-            msg = "PeriodIndex.format is deprecated"
-            with tm.assert_produces_warning(FutureWarning, match=msg):
-                formatted = per.format(date_format="%y é")
-            assert formatted[0] == "03 é"
-            assert formatted[1] == "03 é"
-
-    @pytest.mark.parametrize(
-        "locale_str",
-        [
-            pytest.param(None, id=str(locale.getlocale())),
-            "it_IT.utf8",
-            "it_IT",  # Note: encoding will be 'ISO8859-1'
-            "zh_CN.utf8",
-            "zh_CN",  # Note: encoding will be 'gb2312'
-        ],
-    )
-    def test_period_custom_locale_directive(self, locale_str):
-        # GH#46319 locale-specific directive leads to non-utf8 c strftime char* result
-
-        # Skip if locale cannot be set
-        if locale_str is not None and not tm.can_set_locale(locale_str, locale.LC_ALL):
-            pytest.skip(f"Skipping as locale '{locale_str}' cannot be set on host.")
-
-        # Change locale temporarily for this test.
-        with tm.set_locale(locale_str, locale.LC_ALL) if locale_str else nullcontext():
-            # Get locale-specific reference
-            am_local, pm_local = get_local_am_pm()
-
-            # Scalar
-            per = pd.Period("2018-03-11 13:00", freq="h")
-            assert per.strftime("%p") == pm_local
-
-            # Index
-            per = pd.period_range("2003-01-01 01:00:00", periods=2, freq="12h")
-            msg = "PeriodIndex.format is deprecated"
-            with tm.assert_produces_warning(FutureWarning, match=msg):
-                formatted = per.format(date_format="%y %I:%M:%S%p")
-            assert formatted[0] == f"03 01:00:00{am_local}"
-            assert formatted[1] == f"03 01:00:00{pm_local}"
diff --git a/pandas/tests/indexes/ranges/test_range.py b/pandas/tests/indexes/ranges/test_range.py
index 06e19eeca6766..29d6916cc7f08 100644
--- a/pandas/tests/indexes/ranges/test_range.py
+++ b/pandas/tests/indexes/ranges/test_range.py
@@ -242,11 +242,6 @@ def test_cache(self):
             pass
         assert idx._cache == {}
 
-        msg = "RangeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            idx.format()
-        assert idx._cache == {}
-
         df = pd.DataFrame({"a": range(10)}, index=idx)
 
         # df.__repr__ should not populate index cache
@@ -570,15 +565,6 @@ def test_engineless_lookup(self):
 
         assert "_engine" not in idx._cache
 
-    def test_format_empty(self):
-        # GH35712
-        empty_idx = RangeIndex(0)
-        msg = r"RangeIndex\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format() == []
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format(name=True) == [""]
-
     @pytest.mark.parametrize(
         "ri",
         [
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index 5f08443b4b5b5..948f369edf072 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -686,41 +686,6 @@ def test_is_object(self, index, expected, using_infer_string):
     def test_summary(self, index):
         index._summary()
 
-    def test_format_bug(self):
-        # GH 14626
-        # windows has different precision on datetime.datetime.now (it doesn't
-        # include us since the default for Timestamp shows these but Index
-        # formatting does not we are skipping)
-        now = datetime.now()
-        msg = r"Index\.format is deprecated"
-
-        if not str(now).endswith("000"):
-            index = Index([now])
-            with tm.assert_produces_warning(FutureWarning, match=msg):
-                formatted = index.format()
-            expected = [str(index[0])]
-            assert formatted == expected
-
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            Index([]).format()
-
-    @pytest.mark.parametrize("vals", [[1, 2.0 + 3.0j, 4.0], ["a", "b", "c"]])
-    def test_format_missing(self, vals, nulls_fixture):
-        # 2845
-        vals = list(vals)  # Copy for each iteration
-        vals.append(nulls_fixture)
-        index = Index(vals, dtype=object)
-        # TODO: case with complex dtype?
-
-        msg = r"Index\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            formatted = index.format()
-        null_repr = "NaN" if isinstance(nulls_fixture, float) else str(nulls_fixture)
-        expected = [str(index[0]), str(index[1]), str(index[2]), null_repr]
-
-        assert formatted == expected
-        assert index[3] is nulls_fixture
-
     def test_logical_compat(self, all_boolean_reductions, simple_index):
         index = simple_index
         left = getattr(index, all_boolean_reductions)()
diff --git a/pandas/tests/indexes/test_old_base.py b/pandas/tests/indexes/test_old_base.py
index d944c204f33ec..cc496dbee86d3 100644
--- a/pandas/tests/indexes/test_old_base.py
+++ b/pandas/tests/indexes/test_old_base.py
@@ -567,29 +567,6 @@ def test_equals_op(self, simple_index):
             tm.assert_numpy_array_equal(index_a == item, expected3)
             tm.assert_series_equal(series_a == item, Series(expected3))
 
-    def test_format(self, simple_index):
-        # GH35439
-        if is_numeric_dtype(simple_index.dtype) or isinstance(
-            simple_index, DatetimeIndex
-        ):
-            pytest.skip("Tested elsewhere.")
-        idx = simple_index
-        expected = [str(x) for x in idx]
-        msg = r"Index\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert idx.format() == expected
-
-    def test_format_empty(self, simple_index):
-        # GH35712
-        if isinstance(simple_index, (PeriodIndex, RangeIndex)):
-            pytest.skip("Tested elsewhere")
-        empty_idx = type(simple_index)([])
-        msg = r"Index\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format() == []
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert empty_idx.format(name=True) == [""]
-
     def test_fillna(self, index):
         # GH 11343
         if len(index) == 0:
@@ -927,17 +904,6 @@ def test_view(self, simple_index):
             idx_view = idx.view(index_cls)
         tm.assert_index_equal(idx, index_cls(idx_view, name="Foo"), exact=True)
 
-    def test_format(self, simple_index):
-        # GH35439
-        if isinstance(simple_index, DatetimeIndex):
-            pytest.skip("Tested elsewhere")
-        idx = simple_index
-        max_width = max(len(str(x)) for x in idx)
-        expected = [str(x).ljust(max_width) for x in idx]
-        msg = r"Index\.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            assert idx.format() == expected
-
     def test_insert_non_na(self, simple_index):
         # GH#43921 inserting an element that we know we can hold should
         #  not change dtype or type (except for RangeIndex)
diff --git a/pandas/tests/series/test_formats.py b/pandas/tests/series/test_formats.py
index a1c5018ea7961..1a3b46ec8196a 100644
--- a/pandas/tests/series/test_formats.py
+++ b/pandas/tests/series/test_formats.py
@@ -19,7 +19,6 @@
     period_range,
     timedelta_range,
 )
-import pandas._testing as tm
 
 
 class TestSeriesRepr:
@@ -254,14 +253,6 @@ def test_index_repr_in_frame_with_nan(self):
 
         assert repr(s) == exp
 
-    def test_format_pre_1900_dates(self):
-        rng = date_range("1/1/1850", "1/1/1950", freq="YE-DEC")
-        msg = "DatetimeIndex.format is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            rng.format()
-        ts = Series(1, index=rng)
-        repr(ts)
-
     def test_series_repr_nat(self):
         series = Series([0, 1000, 2000, pd.NaT._value], dtype="M8[ns]")
 
diff --git a/pandas/tests/test_algos.py b/pandas/tests/test_algos.py
index 3fd771c7fe31a..057a5a627370e 100644
--- a/pandas/tests/test_algos.py
+++ b/pandas/tests/test_algos.py
@@ -1212,9 +1212,7 @@ def test_value_counts(self):
         factor = cut(arr, 4)
 
         # assert isinstance(factor, n)
-        msg = "pandas.value_counts is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = algos.value_counts(factor)
+        result = algos.value_counts_internal(factor)
         breaks = [-1.606, -1.018, -0.431, 0.155, 0.741]
         index = IntervalIndex.from_breaks(breaks).astype(CategoricalDtype(ordered=True))
         expected = Series([1, 0, 2, 1], index=index, name="count")
@@ -1222,16 +1220,13 @@ def test_value_counts(self):
 
     def test_value_counts_bins(self):
         s = [1, 2, 3, 4]
-        msg = "pandas.value_counts is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = algos.value_counts(s, bins=1)
+        result = algos.value_counts_internal(s, bins=1)
         expected = Series(
             [4], index=IntervalIndex.from_tuples([(0.996, 4.0)]), name="count"
         )
         tm.assert_series_equal(result, expected)
 
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = algos.value_counts(s, bins=2, sort=False)
+        result = algos.value_counts_internal(s, bins=2, sort=False)
         expected = Series(
             [2, 2],
             index=IntervalIndex.from_tuples([(0.996, 2.5), (2.5, 4.0)]),
@@ -1240,45 +1235,35 @@ def test_value_counts_bins(self):
         tm.assert_series_equal(result, expected)
 
     def test_value_counts_dtypes(self):
-        msg2 = "pandas.value_counts is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg2):
-            result = algos.value_counts(np.array([1, 1.0]))
+        result = algos.value_counts_internal(np.array([1, 1.0]))
         assert len(result) == 1
 
-        with tm.assert_produces_warning(FutureWarning, match=msg2):
-            result = algos.value_counts(np.array([1, 1.0]), bins=1)
+        result = algos.value_counts_internal(np.array([1, 1.0]), bins=1)
         assert len(result) == 1
 
-        with tm.assert_produces_warning(FutureWarning, match=msg2):
-            result = algos.value_counts(Series([1, 1.0, "1"]))  # object
+        result = algos.value_counts_internal(Series([1, 1.0, "1"]))  # object
         assert len(result) == 2
 
         msg = "bins argument only works with numeric data"
         with pytest.raises(TypeError, match=msg):
-            with tm.assert_produces_warning(FutureWarning, match=msg2):
-                algos.value_counts(np.array(["1", 1], dtype=object), bins=1)
+            algos.value_counts_internal(np.array(["1", 1], dtype=object), bins=1)
 
     def test_value_counts_nat(self):
         td = Series([np.timedelta64(10000), NaT], dtype="timedelta64[ns]")
         dt = to_datetime(["NaT", "2014-01-01"])
 
-        msg = "pandas.value_counts is deprecated"
-
         for ser in [td, dt]:
-            with tm.assert_produces_warning(FutureWarning, match=msg):
-                vc = algos.value_counts(ser)
-                vc_with_na = algos.value_counts(ser, dropna=False)
+            vc = algos.value_counts_internal(ser)
+            vc_with_na = algos.value_counts_internal(ser, dropna=False)
             assert len(vc) == 1
             assert len(vc_with_na) == 2
 
         exp_dt = Series({Timestamp("2014-01-01 00:00:00"): 1}, name="count")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result_dt = algos.value_counts(dt)
+        result_dt = algos.value_counts_internal(dt)
         tm.assert_series_equal(result_dt, exp_dt)
 
         exp_td = Series({np.timedelta64(10000): 1}, name="count")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result_td = algos.value_counts(td)
+        result_td = algos.value_counts_internal(td)
         tm.assert_series_equal(result_td, exp_td)
 
     @pytest.mark.parametrize("dtype", [object, "M8[us]"])
@@ -1435,16 +1420,13 @@ def test_value_counts_normalized(self, dtype):
     def test_value_counts_uint64(self):
         arr = np.array([2**63], dtype=np.uint64)
         expected = Series([1], index=[2**63], name="count")
-        msg = "pandas.value_counts is deprecated"
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = algos.value_counts(arr)
+        result = algos.value_counts_internal(arr)
 
         tm.assert_series_equal(result, expected)
 
         arr = np.array([-1, 2**63], dtype=object)
         expected = Series([1, 1], index=[-1, 2**63], name="count")
-        with tm.assert_produces_warning(FutureWarning, match=msg):
-            result = algos.value_counts(arr)
+        result = algos.value_counts_internal(arr)
 
         tm.assert_series_equal(result, expected)