diff --git a/CHANGELOG.md b/CHANGELOG.md
index dff6b1b5c1..12865289ec 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -22,6 +22,8 @@ and this project adheres to [Semantic Versioning](http://semver.org/spec/v2.0.0.
 
 * Corner case optimisation for mapping over `iota` (#1874).
 
+* AD for certain combinations of `map` and indexing (#1878).
+
 ## [0.23.1]
 
 ### Added
diff --git a/src/Futhark/AD/Rev/Map.hs b/src/Futhark/AD/Rev/Map.hs
index 47577266cb..ca6898f684 100644
--- a/src/Futhark/AD/Rev/Map.hs
+++ b/src/Futhark/AD/Rev/Map.hs
@@ -1,5 +1,7 @@
 {-# LANGUAGE TypeFamilies #-}
 
+-- | VJP transformation for Map SOACs.  This is a pretty complicated
+-- case due to the possibility of free variables.
 module Futhark.AD.Rev.Map (vjpMap) where
 
 import Control.Monad
@@ -73,6 +75,8 @@ withAcc inputs m = do
     subAD $ mkLambda (cert_params ++ acc_params) $ m $ map paramName acc_params
   letTupExp "withhacc_res" $ WithAcc inputs acc_lam
 
+-- | Perform VJP on a Map.  The 'Adj' list is the adjoints of the
+-- result of the map.
 vjpMap :: VjpOps -> [Adj] -> StmAux () -> SubExp -> Lambda SOACS -> [VName] -> ADM ()
 vjpMap ops res_adjs _ w map_lam as
   | Just res_ivs <- mapM isSparse res_adjs = returnSweepCode $ do
@@ -80,7 +84,7 @@ vjpMap ops res_adjs _ w map_lam as
       -- (length res_ivs), there is no need for the return sweep code to
       -- contain a Map at all.
 
-      free <- filterM isActive $ namesToList $ freeIn map_lam
+      free <- filterM isActive $ namesToList $ freeIn map_lam `namesSubtract` namesFromList as
       free_ts <- mapM lookupType free
       let adjs_for = map paramName (lambdaParams map_lam) ++ free
           adjs_ts = map paramType (lambdaParams map_lam) ++ free_ts
@@ -96,7 +100,13 @@ vjpMap ops res_adjs _ w map_lam as
             forM_ (zip as adjs_ts) $ \(a, t) -> do
               scratch <- letSubExp "oo_scratch" =<< eBlank t
               updateAdjIndex a (OutOfBounds, adj_i) scratch
-            first subExpsRes . adjsReps <$> mapM lookupAdj as
+            -- We must make sure that all free variables have the same
+            -- representation in the oo-branch as in the ib-branch.
+            -- In practice we do this by manifesting the adjoint.
+            -- This is probably efficient, since the adjoint of a free
+            -- variable is probably either a scalar or an accumulator.
+            forM_ free $ \v -> insAdj v =<< adjVal =<< lookupAdj v
+            first subExpsRes . adjsReps <$> mapM lookupAdj (as <> free)
           inBounds res_i adj_i adj_v = subAD . buildRenamedBody $ do
             forM_ (zip (lambdaParams map_lam) as) $ \(p, a) -> do
               a_t <- lookupType a
@@ -105,15 +115,19 @@ vjpMap ops res_adjs _ w map_lam as
             adj_elems <-
               fmap (map resSubExp) . bodyBind . lambdaBody
                 =<< vjpLambda ops (oneHot res_i (AdjVal adj_v)) adjs_for map_lam
-            forM_ (zip as adj_elems) $ \(a, a_adj_elem) -> do
+            let (as_adj_elems, free_adj_elems) = splitAt (length as) adj_elems
+            forM_ (zip as as_adj_elems) $ \(a, a_adj_elem) ->
               updateAdjIndex a (AssumeBounds, adj_i) a_adj_elem
-            first subExpsRes . adjsReps <$> mapM lookupAdj as
+            forM_ (zip free free_adj_elems) $ \(v, adj_se) -> do
+              adj_se_v <- letExp "adj_v" (BasicOp $ SubExp adj_se)
+              insAdj v adj_se_v
+            first subExpsRes . adjsReps <$> mapM lookupAdj (as <> free)
 
           -- Generate an iteration of the map function for every
           -- position.  This is a bit inefficient - probably we could do
           -- some deduplication.
           forPos res_i (check, adj_i, adj_v) = do
-            as_adj <-
+            adjs <-
               case check of
                 CheckBounds b -> do
                   (obbranch, mkadjs) <- ooBounds adj_i
@@ -129,7 +143,7 @@ vjpMap ops res_adjs _ w map_lam as
                 OutOfBounds ->
                   mapM lookupAdj as
 
-            zipWithM setAdj as as_adj
+            zipWithM setAdj (as <> free) adjs
 
           -- Generate an iteration of the map function for every result.
           forRes res_i = mapM_ (forPos res_i)
diff --git a/tests/ad/map6.fut b/tests/ad/map6.fut
new file mode 100644
index 0000000000..b7a40aafa3
--- /dev/null
+++ b/tests/ad/map6.fut
@@ -0,0 +1,29 @@
+-- #1878
+-- ==
+-- entry: fwd_J rev_J
+-- input { [1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0] }
+-- output { [[0.0, 2.0, 3.0, 4.0],
+--           [0.0, 0.0, 1.0, 1.0],
+--           [0.0, 0.0, 0.0, 1.0],
+--           [0.0, 0.0, 0.0, 0.0],
+--           [-4.0, -6.0, -7.0, -8.0],
+--           [0.0, 0.0, -1.0, -1.0],
+--           [0.0, 0.0, 0.0, -1.0],
+--           [0.0, 0.0, 0.0, 0.0]]
+--        }
+
+def obj (x : [8]f64) =
+  #[unsafe] -- For simplicity of generated code.
+  let col_w_pre_red =
+    tabulate_3d 4 2 4 (\ k i j -> x[k+j]*x[i+j])
+  let col_w_red =
+    map (map f64.sum) col_w_pre_red
+  let col_eq : [4]f64 =
+    map (\w -> w[0] - w[1]) col_w_red
+  in col_eq
+
+entry fwd_J (x : [8]f64)  =
+  tabulate 8 (\i -> jvp obj x (replicate 8 0 with [i] = 1))
+
+entry rev_J (x : [8]f64)  =
+  transpose (tabulate 4 (\i -> vjp obj x (replicate 4 0 with [i] = 1)))
diff --git a/tests/ad/map7.fut b/tests/ad/map7.fut
new file mode 100644
index 0000000000..01f5c3e249
--- /dev/null
+++ b/tests/ad/map7.fut
@@ -0,0 +1,22 @@
+-- #1878.  The interesting thing here is that the sparse adjoint also
+-- has active free variables.
+-- ==
+-- entry: fwd_J rev_J
+-- input { [1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0] }
+-- output { [0.0, 0.0, 0.0, 0.0, -4.0, 0.0, 0.0, 0.0] }
+
+def obj (x : [8]f64) =
+  #[unsafe] -- For simplicity of generated code.
+  let col_w_pre_red =
+    tabulate_3d 4 2 4 (\ k i j -> x[k+j]*x[i+j])
+  let col_w_red =
+    map (map f64.sum) col_w_pre_red
+  let col_eq : [4]f64 =
+    map (\w -> w[0] - w[1]) col_w_red
+  in f64.maximum col_eq
+
+entry fwd_J (x : [8]f64)  =
+  tabulate 8 (\i -> jvp obj x (replicate 8 0 with [i] = 1))
+
+entry rev_J (x : [8]f64)  =
+  vjp obj x 1