diff --git a/src/Control/Optics/Linear/Internal.hs b/src/Control/Optics/Linear/Internal.hs
index d0a532d2..bb159a71 100644
--- a/src/Control/Optics/Linear/Internal.hs
+++ b/src/Control/Optics/Linear/Internal.hs
@@ -13,7 +13,8 @@ module Control.Optics.Linear.Internal
   , Iso, Iso'
   , Lens, Lens'
   , Prism, Prism'
-  , Traversal, Traversal'
+  , PTraversal, PTraversal'
+  , DTraversal, DTraversal'
     -- * Composing optics
   , (.>)
     -- * Common optics
@@ -21,30 +22,38 @@ module Control.Optics.Linear.Internal
   , _1, _2
   , _Left, _Right
   , _Just, _Nothing
-  , traversed
+  , ptraversed, dtraversed
+  , both, both'
     -- * Using optics
   , get, set, gets
+  , get', gets', set', set''
   , match, match', build
+  , preview
   , over, over'
   , traverseOf, traverseOf'
   , lengthOf
-  , withIso
+  , withIso, withLens
+  , toListOf
     -- * Constructing optics
-  , iso, prism
+  , iso, prism, lens
   )
   where
 
 import qualified Control.Arrow as NonLinear
 import qualified Data.Bifunctor.Linear as Bifunctor
 import Data.Bifunctor.Linear (SymmetricMonoidal)
-import Data.Monoid
-import Data.Profunctor.Linear
+import Data.Monoid (First(..), Sum(..))
+import Data.Functor.Const
 import Data.Functor.Linear
+import Data.Profunctor.Linear
 import qualified Data.Profunctor.Kleisli.Linear as Linear
 import Data.Void
 import Prelude.Linear
 import qualified Prelude as P
 
+-- TODO: documentation in this module
+-- Put the functions in some sensible order: possibly split into separate
+-- Lens/Prism/Traversal/Iso modules
 newtype Optic_ arr a b s t = Optical (a `arr` b -> s `arr` t)
 
 type Optic c a b s t =
@@ -56,8 +65,12 @@ type Lens a b s t = Optic (Strong (,) ()) a b s t
 type Lens' a s = Lens a a s s
 type Prism a b s t = Optic (Strong Either Void) a b s t
 type Prism' a s = Prism a a s s
-type Traversal a b s t = Optic Wandering a b s t
-type Traversal' a s = Traversal a a s s
+type PTraversal a b s t = Optic PWandering a b s t
+type PTraversal' a s = PTraversal a a s s
+type DTraversal a b s t = Optic DWandering a b s t
+type DTraversal' a s = DTraversal a a s s
+-- XXX: these will unify into
+-- type Traversal (p :: Multiplicity) a b s t = Optic (Wandering p) a b s t
 
 swap :: SymmetricMonoidal m u => Iso (a `m` b) (c `m` d) (b `m` a) (d `m` c)
 swap = iso Bifunctor.swap Bifunctor.swap
@@ -68,6 +81,12 @@ assoc = iso Bifunctor.lassoc Bifunctor.rassoc
 (.>) :: Optic_ arr a b s t -> Optic_ arr x y a b -> Optic_ arr x y s t
 Optical f .> Optical g = Optical (f P.. g)
 
+lens :: (s ->. (a, b ->. t)) -> Lens a b s t
+lens k = Optical $ \f -> dimap k (\(x,g) -> g $ x) (first f)
+
+withLens :: Optic_ (Linear.Kleisli (OtherFunctor a b)) a b s t -> s ->. (a, b ->. t)
+withLens (Optical l) s = runOtherFunctor (Linear.runKleisli (l (Linear.Kleisli (\a -> OtherFunctor (a, id)))) s)
+
 prism :: (b ->. t) -> (s ->. Either t a) -> Prism a b s t
 prism b s = Optical $ \f -> dimap s (either id id) (second (rmap b f))
 
@@ -77,6 +96,37 @@ _1 = Optical first
 _2 :: Lens a b (c,a) (c,b)
 _2 = Optical second
 
+-- XXX: these will unify to
+-- > both :: forall (p :: Multiplicity). Traversal p a b (a,a) (b,b)
+both' :: PTraversal a b (a,a) (b,b)
+both' = _Pairing .> ptraversed
+
+both :: DTraversal a b (a,a) (b,b)
+both = _Pairing .> dtraversed
+
+-- XXX: these are a special case of Bitraversable, but just the simple case
+-- is included here for now
+_Pairing :: Iso (Pair a) (Pair b) (a,a) (b,b)
+_Pairing = iso Paired unpair
+
+newtype Pair a = Paired (a,a)
+unpair :: Pair a ->. (a,a)
+unpair (Paired x) = x
+
+instance P.Functor Pair where
+  fmap f (Paired (x,y)) = Paired (f x, f y)
+instance Functor Pair where
+  fmap f (Paired (x,y)) = Paired (f x, f y)
+instance Foldable Pair where
+  foldMap f (Paired (x,y)) = f x P.<> f y
+instance P.Traversable Pair where
+  traverse f (Paired (x,y)) = Paired P.<$> ((,) P.<$> f x P.<*> f y)
+instance Traversable Pair where
+  traverse f (Paired (x,y)) = Paired <$> ((,) <$> f x <*> f y)
+
+toListOf :: Optic_ (NonLinear.Kleisli (Const [a])) a b s t -> s -> [a]
+toListOf l = gets l (\a -> [a])
+
 _Left :: Prism a b (Either a c) (Either b c)
 _Left = Optical first
 
@@ -89,8 +139,11 @@ _Just = prism Just (maybe (Left Nothing) Right)
 _Nothing :: Prism' () (Maybe a)
 _Nothing = prism (\() -> Nothing) Left
 
-traversed :: Traversable t => Traversal a b (t a) (t b)
-traversed = Optical wander
+ptraversed :: P.Traversable t => PTraversal a b (t a) (t b)
+ptraversed = Optical pwander
+
+dtraversed :: Traversable t => DTraversal a b (t a) (t b)
+dtraversed = Optical dwander
 
 over :: Optic_ LinearArrow a b s t -> (a ->. b) -> s ->. t
 over (Optical l) f = getLA (l (LA f))
@@ -104,6 +157,21 @@ get l = gets l P.id
 gets :: Optic_ (NonLinear.Kleisli (Const r)) a b s t -> (a -> r) -> s -> r
 gets (Optical l) f s = getConst' (NonLinear.runKleisli (l (NonLinear.Kleisli (Const P.. f))) s)
 
+preview :: Optic_ (NonLinear.Kleisli (Const (First a))) a b s t -> s -> Maybe a
+preview (Optical l) s = getFirst (getConst (NonLinear.runKleisli (l (NonLinear.Kleisli (\a -> Const (First (Just a))))) s))
+
+get' :: Optic_ (Linear.Kleisli (Const (Top, a))) a b s t -> s ->. (Top, a)
+get' l = gets' l id
+
+gets' :: Optic_ (Linear.Kleisli (Const (Top, r))) a b s t -> (a ->. r) -> s ->. (Top, r)
+gets' (Optical l) f s = getConst' (Linear.runKleisli (l (Linear.Kleisli (\a -> Const (mempty, f a)))) s)
+
+set' :: Optic_ (Linear.Kleisli (MyFunctor a b)) a b s t -> s ->. b ->. (a, t)
+set' (Optical l) s = runMyFunctor (Linear.runKleisli (l (Linear.Kleisli (\a -> MyFunctor (\b -> (a,b))))) s)
+
+set'' :: Optic_ (NonLinear.Kleisli (Reader b)) a b s t -> b ->. s -> t
+set'' (Optical l) b s = runReader (NonLinear.runKleisli (l (NonLinear.Kleisli (const (Reader id)))) s) b
+
 set :: Optic_ (->) a b s t -> b -> s -> t
 set (Optical l) x = l (const x)
 
diff --git a/src/Data/Monoid/Linear.hs b/src/Data/Monoid/Linear.hs
index 1cadea9a..9fcdecda 100644
--- a/src/Data/Monoid/Linear.hs
+++ b/src/Data/Monoid/Linear.hs
@@ -3,6 +3,7 @@
 {-# LANGUAGE GeneralizedNewtypeDeriving #-}
 {-# LANGUAGE LinearTypes #-}
 {-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE ExistentialQuantification #-}
 {-# LANGUAGE StandaloneDeriving #-}
 
 -- | = The linear monoid hierarchy
@@ -14,6 +15,7 @@ module Data.Monoid.Linear
   , Monoid(..)
   , Endo(..), appEndo
   , NonLinear(..)
+  , Top, throw
   , module Data.Semigroup
   )
   where
@@ -80,3 +82,15 @@ newtype NonLinear a = NonLinear a
 
 instance Semigroup a => Prelude.Semigroup (NonLinear a) where
   NonLinear a <> NonLinear b = NonLinear (a <> b)
+
+data Top = forall x. Top x
+throw :: x ->. Top
+throw = Top
+
+instance Prelude.Semigroup Top where
+  Top x <> Top y = Top (x,y)
+instance Semigroup Top where
+  Top x <> Top y = Top (x,y)
+instance Prelude.Monoid Top where
+  mempty = Top ()
+instance Monoid Top where
diff --git a/src/Data/Profunctor/Kleisli/Linear.hs b/src/Data/Profunctor/Kleisli/Linear.hs
index 61951ef5..6394c788 100644
--- a/src/Data/Profunctor/Kleisli/Linear.hs
+++ b/src/Data/Profunctor/Kleisli/Linear.hs
@@ -41,22 +41,14 @@ instance Control.Applicative f => Strong Either Void (Kleisli f) where
   first  (Kleisli f) = Kleisli (either (Data.fmap Left . f) (Control.pure . Right))
   second (Kleisli g) = Kleisli (either (Control.pure . Left) (Data.fmap Right . g))
 
-instance Control.Applicative f => Wandering (Kleisli f) where
-  wander (Kleisli f) = Kleisli (Data.traverse f)
+instance Control.Applicative f => DWandering (Kleisli f) where
+  dwander (Kleisli f) = Kleisli (Data.traverse f)
 
 -- | Linear co-Kleisli arrows for the comonad `w`. These arrows are still
 -- useful in the case where `w` is not a comonad however, and some
 -- profunctorial properties still hold in this weaker setting.
 -- However stronger requirements on `f` are needed for profunctorial
 -- strength, so we have fewer instances.
---
--- Category theoretic remark: duality doesn't work in the obvious way, since
--- (,) isn't the categorical product. Instead, we have a product (&), called
--- "With", defined by
--- > type With a b = forall r. Either (a ->. r) (b ->. r) ->. r
--- which satisfies the universal property of the product of `a` and `b`.
--- CoKleisli arrows are strong with respect to this monoidal structure,
--- although this might not be useful...
 newtype CoKleisli w a b = CoKleisli { runCoKleisli :: w a ->. b }
 
 instance Data.Functor f => Profunctor (CoKleisli f) where
diff --git a/src/Data/Profunctor/Linear.hs b/src/Data/Profunctor/Linear.hs
index e71493d0..82582196 100644
--- a/src/Data/Profunctor/Linear.hs
+++ b/src/Data/Profunctor/Linear.hs
@@ -1,27 +1,39 @@
+{-# LANGUAGE GADTs #-}
 {-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE KindSignatures #-}
 {-# LANGUAGE LambdaCase #-}
 {-# LANGUAGE LinearTypes #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
 {-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RankNTypes #-}
 {-# LANGUAGE TupleSections #-}
 {-# LANGUAGE TypeOperators #-}
 
+{-# OPTIONS_GHC -fno-warn-orphans #-}
+
 module Data.Profunctor.Linear
   ( Profunctor(..)
   , Monoidal(..)
   , Strong(..)
-  , Wandering(..)
+  , PWandering(..)
+  , DWandering(..)
   , LinearArrow(..), getLA
   , Exchange(..)
+  , Top
+  , MyFunctor(..), runMyFunctor
+  , OtherFunctor(..), runOtherFunctor
   ) where
 
 import qualified Data.Functor.Linear as Data
+import qualified Control.Monad.Linear as Control
 import Data.Bifunctor.Linear hiding (first, second)
 import Prelude.Linear
 import Data.Void
 import qualified Prelude
 import Control.Arrow (Kleisli(..))
+import Data.Monoid.Linear
+import Data.Functor.Const
 
 -- TODO: write laws
 
@@ -55,8 +67,17 @@ class (SymmetricMonoidal m u, Profunctor arr) => Strong m u arr where
   second arr = dimap swap swap (first arr)
   {-# INLINE second #-}
 
-class (Strong (,) () arr, Strong Either Void arr) => Wandering arr where
-  wander :: Data.Traversable f => a `arr` b -> f a `arr` f b
+-- XXX: Just as Prelude.Functor/Data.Functor will combine into
+-- > `class Functor (p :: Multiplicity) f`
+-- so will Traversable, and then we would instead write
+-- > class (...) => Wandering (p :: Multiplicity) arr where
+-- >   wander :: Traversable p f => a `arr` b -> f a `arr` f b
+-- For now, however, we cannot do this, so we use two classes instead:
+-- PreludeWandering and DataWandering
+class (Strong (,) () arr, Strong Either Void arr) => PWandering arr where
+  pwander :: Prelude.Traversable f => a `arr` b -> f a `arr` f b
+class (Strong (,) () arr, Strong Either Void arr) => DWandering arr where
+  dwander :: Data.Traversable f => a `arr` b -> f a `arr` f b
 
 ---------------
 -- Instances --
@@ -78,6 +99,9 @@ instance Strong Either Void LinearArrow where
   first  (LA f) = LA $ either (Left . f) Right
   second (LA g) = LA $ either Left (Right . g)
 
+instance DWandering LinearArrow where
+  dwander (LA f) = LA (Data.fmap f)
+
 instance Profunctor (->) where
   dimap f g h x = g (h (f x))
 instance Strong (,) () (->) where
@@ -85,6 +109,8 @@ instance Strong (,) () (->) where
 instance Strong Either Void (->) where
   first f (Left x) = Left (f x)
   first _ (Right y) = Right y
+instance PWandering (->) where
+  pwander = Prelude.fmap
 
 data Exchange a b s t = Exchange (s ->. a) (b ->. t)
 instance Profunctor (Exchange a b) where
@@ -101,3 +127,29 @@ instance Prelude.Applicative f => Strong Either Void (Kleisli f) where
   first  (Kleisli f) = Kleisli $ \case
                                    Left  x -> Prelude.fmap Left (f x)
                                    Right y -> Prelude.pure (Right y)
+
+instance Control.Functor (Const (Top, a)) where
+  fmap f (Const (t, x)) = Const (throw f <> t, x)
+instance Monoid a => Control.Applicative (Const (Top, a)) where
+  pure x = Const (throw x, mempty)
+  Const x <*> Const y = Const (x <> y)
+
+-- TODO: pick a more sensible name for this
+newtype MyFunctor a b t = MyFunctor (b ->. (a, t))
+runMyFunctor :: MyFunctor a b t ->. b ->. (a, t)
+runMyFunctor (MyFunctor f) = f
+
+instance Data.Functor (MyFunctor a b) where
+  fmap f (MyFunctor g) = MyFunctor (getLA (second (LA f)) . g)
+instance Control.Functor (MyFunctor a b) where
+  fmap f (MyFunctor g) = MyFunctor (thing f . g)
+    where thing :: (c ->. d) ->. (e, c) ->. (e, d)
+          thing k (x,y) = (x, k y)
+
+newtype OtherFunctor a b t = OtherFunctor (a, b ->. t)
+runOtherFunctor :: OtherFunctor a b t ->. (a, b ->. t)
+runOtherFunctor (OtherFunctor f) = f
+instance Data.Functor (OtherFunctor a b) where
+  fmap f (OtherFunctor (a,g)) = OtherFunctor (a,f . g)
+instance Control.Functor (OtherFunctor a b) where
+  fmap f (OtherFunctor (a,g)) = OtherFunctor (a,f . g)
diff --git a/src/Foreign/Marshal/Pure.hs b/src/Foreign/Marshal/Pure.hs
index 648085c0..0dbd6416 100644
--- a/src/Foreign/Marshal/Pure.hs
+++ b/src/Foreign/Marshal/Pure.hs
@@ -60,7 +60,7 @@ import Foreign.Marshal.Utils
 import Foreign.Ptr
 import Foreign.Storable
 import Foreign.Storable.Tuple ()
-import Prelude (($), return, (<*>))
+import Prelude (($), return, (<*>), (<$>))
 import Prelude.Linear hiding (($))
 import System.IO.Unsafe
 import qualified Unsafe.Linear as Unsafe
diff --git a/src/Prelude/Linear.hs b/src/Prelude/Linear.hs
index 8b18498e..abcf95ec 100644
--- a/src/Prelude/Linear.hs
+++ b/src/Prelude/Linear.hs
@@ -53,7 +53,7 @@ import Prelude hiding
   , foldr
   , maybe
   , (.)
-  , Functor(..)
+  , Functor(..), (<$>)
   , Applicative(..)
   , Monad(..)
   , Traversable(..)
diff --git a/src/System/IO/Linear.hs b/src/System/IO/Linear.hs
index bf6ee2fc..453b6540 100644
--- a/src/System/IO/Linear.hs
+++ b/src/System/IO/Linear.hs
@@ -41,6 +41,7 @@ import qualified Control.Monad.Linear as Control
 import qualified Data.Functor.Linear as Data
 import GHC.Exts (State#, RealWorld)
 import Prelude.Linear hiding (IO)
+import Prelude ((<$>))
 import qualified Unsafe.Linear as Unsafe
 import qualified System.IO as System