diff --git a/src/lib/AbstractSyntax.hs b/src/lib/AbstractSyntax.hs
index 99c922a06..428fa40cc 100644
--- a/src/lib/AbstractSyntax.hs
+++ b/src/lib/AbstractSyntax.hs
@@ -494,6 +494,10 @@ expr = propagateSrcE expr' where
         UApp (mkApp (ns $ fromString rangeName) (ns UHole)) lim
   expr' (CLambda args body) =
     dropSrcE <$> liftM2 buildLam (concat <$> mapM argument args) (block body)
+  expr' (CMap fun array) = do
+    fun' <- expr fun
+    array' <- expr array
+    return $ UMap fun' array'
   expr' (CFor KView indices body) =
     dropSrcE <$> (buildTabLam <$> mapM patOptAnn indices <*> block body)
   expr' (CFor kind indices body) = do
diff --git a/src/lib/CheckType.hs b/src/lib/CheckType.hs
index d72d1712a..5f4c0aca8 100644
--- a/src/lib/CheckType.hs
+++ b/src/lib/CheckType.hs
@@ -777,6 +777,14 @@ typeCheckPrimOp op = case op of
 
 typeCheckPrimHof :: Typer m => PrimHof (Atom i) -> m i o (Type o)
 typeCheckPrimHof hof = addContext ("Checking HOF:\n" ++ pprint hof) case hof of
+  Map fun array -> do
+    Pi (PiType (PiBinder b argTy PlainArrow) Pure resEltTy) <- getTypeE fun
+    let resEltTy' = ignoreHoistFailure $ hoist b resEltTy
+    TabPi (TabPiType binder argEltTy) <- getTypeE array
+    let argEltTy' = ignoreHoistFailure $ hoist binder argEltTy
+    checkAlphaEq argTy argEltTy'
+    refreshAbs (Abs binder UnitE) \binder' _ ->
+      return $ TabPi $ TabPiType binder' (sink resEltTy')
   For _ ixDict f -> do
     ixTy <- ixTyFromDict =<< substM ixDict
     Pi (PiType (PiBinder b argTy PlainArrow) eff eltTy) <- getTypeE f
diff --git a/src/lib/ConcreteSyntax.hs b/src/lib/ConcreteSyntax.hs
index 446ad5d53..60cef64d7 100644
--- a/src/lib/ConcreteSyntax.hs
+++ b/src/lib/ConcreteSyntax.hs
@@ -114,6 +114,7 @@ data Group'
   | CPrefix SourceName Group -- covers unary - and unary + among others
   | CPostfix SourceName Group
   | CLambda [Group] CBlock  -- The arguments do not have Juxtapose at the top level
+  | CMap Group Group -- unary fun, array
   | CFor ForKind [Group] CBlock -- also for_, rof, rof_, view
   | CCase Group [(Group, CBlock)] -- scrutinee, alternatives
   | CIf Group CBlock (Maybe CBlock)
@@ -559,6 +560,14 @@ cLam = do
   body <- cBlock
   return $ CLambda bs body
 
+cMap :: Parser Group'
+cMap = do
+  keyWord MapKW
+  fun <- cGroupNoJuxt
+  keyWord OverKW
+  array <- cGroup
+  return $ CMap fun array
+
 cFor :: Parser Group'
 cFor = do
   kw <- forKW
@@ -704,6 +713,7 @@ leafGroupNoBrackets = do
     _ | isDigit next -> (    CNat   <$> natLit
                          <|> CFloat <$> doubleLit)
     '\\' -> cLam
+    'm'  -> cMap <|> CIdentifier <$> anyName
     -- For exprs include view, for, rof, for_, rof_
     'v'  -> cFor  <|> CIdentifier <$> anyName
     'f'  -> cFor  <|> CIdentifier <$> anyName
diff --git a/src/lib/Imp.hs b/src/lib/Imp.hs
index d3c0fbcac..82c5831f5 100644
--- a/src/lib/Imp.hs
+++ b/src/lib/Imp.hs
@@ -560,6 +560,19 @@ toImpHof :: Emits o => Maybe (Dest o) -> PrimHof (Atom i) -> SubstImpM i o (Atom
 toImpHof maybeDest hof = do
   resultTy <- getTypeSubst (Hof hof)
   case hof of
+    Map (Lam (LamExpr b body)) array -> do
+      rDest <- allocDest maybeDest resultTy
+      TabPi (TabPiType (_:>ixTy) _) <- getTypeSubst array
+      array' <- substM array
+      n <- indexSetSizeImp ixTy
+      emitLoop noHint Fwd n \i -> do
+        idx <- unsafeFromOrdinalImp (sink ixTy) i
+        ithArg <- dropSubst $ translateExpr Nothing $
+                    TabApp (sink array') $ idx :| []
+        ithDest <- destGet (sink rDest) idx
+        void $ extendSubst (b @> SubstVal ithArg) $
+          translateBlock (Just ithDest) body
+      destToAtom rDest
     For d ixDict (Lam (LamExpr b body)) -> do
       ixTy <- ixTyFromDict =<< substM ixDict
       n <- indexSetSizeImp ixTy
diff --git a/src/lib/Inference.hs b/src/lib/Inference.hs
index 028683a6d..2d0f671ba 100644
--- a/src/lib/Inference.hs
+++ b/src/lib/Inference.hs
@@ -918,6 +918,20 @@ getImplicitArg (PiBinder _ argTy arr) = case arr of
     return $ Just $ Con $ DictHole (AlwaysEqual ctx) argTy
   _ -> return Nothing
 
+etaExpand :: EmitsInf n => Atom n -> InfererM i n (Atom n)
+etaExpand fun = do
+  ty <- getType fun
+  case ty of
+    Pi (PiType (PiBinder b argTy arr) eff _) -> do
+      case fun of
+        Lam _ -> pure fun
+        _ -> buildLamInf noHint arr argTy
+              (\b' -> applySubst (b @> b') eff)
+              (\x -> do
+                Distinct <- getDistinct
+                app (sink fun) (Var x))
+    _ -> error "atom must have pi type"
+
 checkOrInferRho :: forall i o. EmitsBoth o
                 => UExpr i -> RequiredTy RhoType o -> InfererM i o (Atom o)
 checkOrInferRho (WithSrcE pos expr) reqTy = do
@@ -943,6 +957,23 @@ checkOrInferRho (WithSrcE pos expr) reqTy = do
       Infer   -> inferULam Pure uLamExpr
     ixTy <- asIxType ty'
     matchRequirement $ TabLam $ TabLamExpr (b':>ixTy) body'
+  UMap fun array -> do
+    array' <- inferRho array
+    arrayTy <- getType array'
+    case arrayTy of
+      TabPi (TabPiType (b:>_) argElemTy) -> do
+        argElemTy' <- case hoist b argElemTy of
+          HoistSuccess ty -> return ty
+          HoistFailure _ -> throw TypeErr "expected non-dependent array type"
+        resElemVar <- liftM Var $ freshInferenceName (TC TypeKind)
+        funTy <- naryNonDepPiType PlainArrow Pure [argElemTy'] resElemVar
+        fun' <- checkOrInferRho fun (Check funTy)
+        -- Eta-expand `fun'` into a `Lam`. Later on we make use of the invariant
+        -- that the first argument of `Map` is a `Lam`.
+        fun'' <- etaExpand fun'
+        result <- liftM Var $ emit $ Hof $ Map fun'' array'
+        matchRequirement result
+      _ -> throw TypeErr "expected array type"
   UFor dir (UForExpr b body) -> do
     allowedEff <- getAllowedEffects
     let uLamExpr = ULamExpr PlainArrow b body
diff --git a/src/lib/Lexing.hs b/src/lib/Lexing.hs
index bf2093f87..2d2b52a28 100644
--- a/src/lib/Lexing.hs
+++ b/src/lib/Lexing.hs
@@ -69,11 +69,14 @@ data KeyWord = DefKW | ForKW | For_KW | RofKW | Rof_KW | CaseKW | OfKW
              | ViewKW | ImportKW | ForeignKW | NamedInstanceKW
              | EffectKW | HandlerKW | JmpKW | CtlKW | ReturnKW | ResumeKW
              | CustomLinearizationKW | CustomLinearizationSymbolicKW
+             | MapKW | OverKW
   deriving (Enum)
 
 keyWordToken :: KeyWord -> String
 keyWordToken = \case
   DefKW           -> "def"
+  MapKW           -> "map_"
+  OverKW          -> "over_"
   ForKW           -> "for"
   RofKW           -> "rof"
   For_KW          -> "for_"
diff --git a/src/lib/PPrint.hs b/src/lib/PPrint.hs
index d3d6e1654..34c6843ec 100644
--- a/src/lib/PPrint.hs
+++ b/src/lib/PPrint.hs
@@ -662,6 +662,8 @@ instance PrettyPrec (UExpr' n) where
                              <+> nest 2 (pLowest body)
       where kw = case dir of Fwd -> "for"
                              Rev -> "rof"
+    UMap fun array -> atPrec LowestPrec $ "map_"  <+> nest 2 (pLowest fun)
+                                      <+> "over_" <+> nest 2 (pLowest array)
     UPi piType -> prettyPrec piType
     UTabPi piType -> prettyPrec piType
     UDecl declExpr -> prettyPrec declExpr
diff --git a/src/lib/QueryType.hs b/src/lib/QueryType.hs
index 303835a0b..4e5c6febf 100644
--- a/src/lib/QueryType.hs
+++ b/src/lib/QueryType.hs
@@ -698,6 +698,12 @@ getTypePrimHof hof = addContext ("Checking HOF:\n" ++ pprint hof) case hof of
     Pi (PiType (PiBinder b _ _) _ eltTy) <- getTypeE f
     ixTy <- ixTyFromDict =<< substM dict
     return $ TabTy (b:>ixTy) eltTy
+  Map fun array -> do
+    Pi (PiType (PiBinder b _ _) _ resEltTy) <- getTypeE fun
+    let resEltTy' = ignoreHoistFailure $ hoist b resEltTy
+    TabPi (TabPiType binder _) <- getTypeE array
+    refreshAbs (Abs binder UnitE) \binder' _ ->
+      return $ TabPi $ TabPiType binder' (sink resEltTy')
   While _ -> return UnitTy
   Linearize f -> do
     Pi (PiType (PiBinder binder a PlainArrow) Pure b) <- getTypeE f
@@ -797,6 +803,7 @@ exprEffects expr = case expr of
     _ -> return Pure
   Hof hof -> case hof of
     For _ _ f     -> functionEffs f
+    Map _ _       -> return Pure
     While body    -> functionEffs body
     Linearize _   -> return Pure  -- Body has to be a pure function
     Transpose _   -> return Pure  -- Body has to be a pure function
diff --git a/src/lib/Simplify.hs b/src/lib/Simplify.hs
index 3bf96a4c3..d59ee255d 100644
--- a/src/lib/Simplify.hs
+++ b/src/lib/Simplify.hs
@@ -835,6 +835,19 @@ projectDictMethod d i = do
 
 simplifyHof :: Emits o => Hof i -> SimplifyM i o (Atom o)
 simplifyHof hof = case hof of
+  Map fun array -> do
+    (fun', Abs b recon) <- simplifyLam fun
+    array' <- simplifyAtom array
+    ans <- liftM Var $ emit $ Hof $ Map fun' array'
+    case recon of
+      IdentityRecon -> return ans
+      LamRecon reconAbs -> do
+        TabPi (TabPiType (_:>ixTy) _) <- getType array'
+        buildTabLam noHint ixTy \i -> do
+          locals <- tabApp (sink ans) $ Var i
+          ithArg <- emitAtomToName =<< (tabApp (sink array') $ Var i)
+          reconAbs' <- applySubst (b @> ithArg) reconAbs
+          applyReconAbs reconAbs' locals
   For d ixDict lam -> do
     ixTy@(IxType _ ixDict') <- ixTyFromDict =<< substM ixDict
     (lam', Abs b recon) <- simplifyLam lam
diff --git a/src/lib/SourceRename.hs b/src/lib/SourceRename.hs
index b7a465fff..7dde587c6 100644
--- a/src/lib/SourceRename.hs
+++ b/src/lib/SourceRename.hs
@@ -207,6 +207,8 @@ instance SourceRenamableE UExpr' where
     UDecl (UDeclExpr decl rest) ->
       sourceRenameB decl \decl' ->
         UDecl <$> UDeclExpr decl' <$> sourceRenameE rest
+    UMap fun array -> UMap <$> sourceRenameE fun
+                           <*> sourceRenameE array
     UFor d (UForExpr pat body) ->
       sourceRenameB pat \pat' ->
         UFor d <$> UForExpr pat' <$> sourceRenameE body
diff --git a/src/lib/Types/Primitives.hs b/src/lib/Types/Primitives.hs
index fada8281a..948bb82c9 100644
--- a/src/lib/Types/Primitives.hs
+++ b/src/lib/Types/Primitives.hs
@@ -158,6 +158,7 @@ traversePrimOp = inline traverse
 
 data PrimHof e =
         For ForAnn e e        -- ix dict, body lambda
+      | Map e e               -- lambda, array
       | While e
       | RunReader e e
       | RunWriter (Maybe e) (BaseMonoidP e) e
diff --git a/src/lib/Types/Source.hs b/src/lib/Types/Source.hs
index d18b6a454..f42fc581a 100644
--- a/src/lib/Types/Source.hs
+++ b/src/lib/Types/Source.hs
@@ -93,6 +93,7 @@ data UExpr' (n::S) =
  | UTabPi  (UTabPiExpr n)
  | UTabApp (UExpr n) (UExpr n)
  | UDecl (UDeclExpr n)
+ | UMap (UExpr n) (UExpr n)
  | UFor Direction (UForExpr n)
  | UCase (UExpr n) [UAlt n]
  | UHole
diff --git a/tests/mymap.dx b/tests/mymap.dx
new file mode 100644
index 000000000..d5fc48c64
--- /dev/null
+++ b/tests/mymap.dx
@@ -0,0 +1,22 @@
+def my_map {a:Type} {b:Type} {n:Type} [Ix n]
+    (f:a -> b)
+    (x:n => a) : n => b =
+  for i:n. f x.i
+
+x0 = [1, 2, 3, 4, 5]
+
+my_map (\x. x+x) x0
+map_ (\x. x+x) over_ x0
+
+my_map (\x. 2*x) x0
+map_ (\x. 2*x) over_ x0
+
+my_map (\x. 2*x) (x0 + x0)
+map_ (\x. 2*x) over_ (x0 + x0)
+-- The following is also parsed as `map_ (\x. 2*x) over_ (x0 + x0)` ... not
+-- intentionally so.
+map_ (\x. 2*x) over_ x0 + x0
+
+my_map (\x. 2*x) x0 + x0
+(map_ (\x. 2*x) over_ x0) + x0
+