diff --git a/src/ChainRules.jl b/src/ChainRules.jl
index cf812063b..4aeb988bf 100644
--- a/src/ChainRules.jl
+++ b/src/ChainRules.jl
@@ -13,6 +13,7 @@ include("rules.jl")
 include("rules/base.jl")
 include("rules/array.jl")
 include("rules/broadcast.jl")
+include("rules/mapreduce.jl")
 include("rules/linalg/utils.jl")
 include("rules/linalg/blas.jl")
 include("rules/linalg/dense.jl")
diff --git a/src/rules/linalg/dense.jl b/src/rules/linalg/dense.jl
index 6f86d25a3..9eb3ee168 100644
--- a/src/rules/linalg/dense.jl
+++ b/src/rules/linalg/dense.jl
@@ -4,14 +4,6 @@ using LinearAlgebra: AbstractTriangular
 # these we can use simpler definitions for `/` and `\`.
 const SquareMatrix{T} = Union{Diagonal{T},AbstractTriangular{T}}
 
-#####
-##### `sum`
-#####
-
-frule(::typeof(sum), x) = (sum(x), Rule(sum))
-
-rrule(::typeof(sum), x) = (sum(x), Rule(cast))
-
 #####
 ##### `dot`
 #####
diff --git a/src/rules/mapreduce.jl b/src/rules/mapreduce.jl
new file mode 100644
index 000000000..25bcfd3fa
--- /dev/null
+++ b/src/rules/mapreduce.jl
@@ -0,0 +1,29 @@
+#####
+##### `map`
+#####
+
+function rrule(::typeof(map), f, xs...)
+    y = map(f, xs...)
+    ∂xs = ntuple(length(xs)) do i
+        Rule() do ȳ
+            map(ȳ, xs...) do ȳi, xis...
+                r = rrule(f, xis...)
+                if r === nothing
+                    throw(ArgumentError("can't differentiate `map` with `$f`; no `rrule` " *
+                                        "is defined for `$f$xis`"))
+                end
+                _, ∂xis = r
+                extern(∂xis[i](ȳi))
+            end
+        end
+    end
+    return y, (DNERule(), ∂xs...)
+end
+
+#####
+##### `sum`
+#####
+
+frule(::typeof(sum), x) = (sum(x), Rule(sum))
+
+rrule(::typeof(sum), x) = (sum(x), Rule(cast))
diff --git a/test/rules/mapreduce.jl b/test/rules/mapreduce.jl
new file mode 100644
index 000000000..6d88b8c30
--- /dev/null
+++ b/test/rules/mapreduce.jl
@@ -0,0 +1,11 @@
+@testset "Maps and Reductions" begin
+    @testset "map" begin
+        rng = MersenneTwister(42)
+        n = 10
+        x = randn(rng, n)
+        vx = randn(rng, n)
+        ȳ = randn(rng, n)
+        rrule_test(map, ȳ, (sin, nothing), (x, vx))
+        rrule_test(map, ȳ, (+, nothing), (x, vx), (randn(rng, n), randn(rng, n)))
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 3f474becc..df2281c13 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -4,7 +4,7 @@ using ChainRules, Test, FDM, LinearAlgebra, Random
 using ChainRules: extern, accumulate, accumulate!, store!, @scalar_rule,
     Wirtinger, wirtinger_primal, wirtinger_conjugate, add_wirtinger, mul_wirtinger,
     Zero, add_zero, mul_zero, One, add_one, mul_one, Casted, cast, add_casted, mul_casted,
-    DNE, Thunk, Casted
+    DNE, Thunk, Casted, DNERule
 using Base.Broadcast: broadcastable
 
 include("test_util.jl")
@@ -15,6 +15,7 @@ include("test_util.jl")
     @testset "rules" begin
         include(joinpath("rules", "base.jl"))
         include(joinpath("rules", "array.jl"))
+        include(joinpath("rules", "mapreduce.jl"))
         @testset "linalg" begin
             include(joinpath("rules", "linalg", "dense.jl"))
             include(joinpath("rules", "linalg", "structured.jl"))
diff --git a/test/test_util.jl b/test/test_util.jl
index a45c121f2..f6cd9992a 100644
--- a/test/test_util.jl
+++ b/test/test_util.jl
@@ -50,23 +50,58 @@ function rrule_test(f, ȳ, (x, x̄)::Tuple{Any, Any}; rtol=1e-9, atol=1e-9, fdm
     test_accumulation(Zero(), dx, ȳ, x̄_ad)
 end
 
+function _make_fdm_call(fdm, f, ȳ, xs, ignores)
+    sig = Expr(:tuple)
+    call = Expr(:call, f)
+    newxs = Any[]
+    arginds = Int[]
+    i = 1
+    for (x, ignore) in zip(xs, ignores)
+        if ignore
+            push!(call.args, x)
+        else
+            push!(call.args, Symbol(:x, i))
+            push!(sig.args, Symbol(:x, i))
+            push!(newxs, x)
+            push!(arginds, i)
+        end
+        i += 1
+    end
+    fdexpr = :(j′vp($fdm, $sig -> $call, $ȳ, $(newxs...)))
+    fd = eval(fdexpr)
+    fd isa Tuple || (fd = (fd,))
+    args = Any[nothing for _ in 1:length(xs)]
+    for (dx, ind) in zip(fd, arginds)
+        args[ind] = dx
+    end
+    return (args...,)
+end
+
 function rrule_test(f, ȳ, xx̄s::Tuple{Any, Any}...; rtol=1e-9, atol=1e-9, fdm=_fdm, kwargs...)
     # Check correctness of evaluation.
     xs, x̄s = collect(zip(xx̄s...))
-    Ω, Δx_rules = ChainRules.rrule(f, xs...)
-    @test f(xs...) == Ω
+    y, rules = rrule(f, xs...)
+    @test f(xs...) == y
 
     # Correctness testing via finite differencing.
-    Δxs_ad = map(Δx_rule->Δx_rule(ȳ), Δx_rules)
-    Δxs_fd = j′vp(fdm, f, ȳ, xs...)
-    for (Δx_ad, Δx_fd) in zip(Δxs_ad, Δxs_fd)
-        @test isapprox(Δx_ad, Δx_fd; rtol=rtol, atol=atol, kwargs...)
+    x̄s_ad = map(rules) do rule
+        rule isa DNERule ? DNE() : rule(ȳ)
+    end
+    x̄s_fd = _make_fdm_call(fdm, f, ȳ, xs, x̄s .== nothing)
+    for (x̄_ad, x̄_fd) in zip(x̄s_ad, x̄s_fd)
+        if x̄_fd === nothing
+            # The way we've structured the above, this tests that the rule is a DNERule
+            @test x̄_ad isa DNE
+        else
+            @test isapprox(x̄_ad, x̄_fd; rtol=rtol, atol=atol, kwargs...)
+        end
     end
 
     # Assuming the above to be correct, check that other ChainRules mechanisms are correct.
-    for (x̄, Δx_rule, Δx_ad) in zip(x̄s, Δx_rules, Δxs_ad)
-        test_accumulation(x̄, Δx_rule, ȳ, Δx_ad)
-        test_accumulation(Zero(), Δx_rule, ȳ, Δx_ad)
+    for (x̄, rule, x̄_ad) in zip(x̄s, rules, x̄s_ad)
+        x̄ === nothing && continue
+        test_accumulation(x̄, rule, ȳ, x̄_ad)
+        test_accumulation(Zero(), rule, ȳ, x̄_ad)
     end
 end