diff --git a/src/mapreduce.jl b/src/mapreduce.jl
index 0595a3b8..b1c95636 100644
--- a/src/mapreduce.jl
+++ b/src/mapreduce.jl
@@ -285,3 +285,53 @@ end
         @inbounds return similar_type(a, T, Size($Snew))(tuple($(exprs...)))
     end
 end
+
+struct _InitialValue end
+
+_maybe_val(dims::Integer) = Val(Int(dims))
+_maybe_val(dims) = dims
+_valof(::Val{D}) where D = D
+
+@inline Base.accumulate(op::F, a::StaticVector; dims = :, init = _InitialValue()) where {F} =
+    _accumulate(op, a, _maybe_val(dims), init)
+
+@inline Base.accumulate(op::F, a::StaticArray; dims, init = _InitialValue()) where {F} =
+    _accumulate(op, a, _maybe_val(dims), init)
+
+@inline function _accumulate(op::F, a::StaticArray, dims::Union{Val,Colon}, init) where {F}
+    # Adjoin the initial value to `op`:
+    rf(x, y) = x isa _InitialValue ? Base.reduce_first(op, y) : op(x, y)
+
+    if isempty(a)
+        T = return_type(rf, Tuple{typeof(init), eltype(a)})
+        return similar_type(a, T)()
+    end
+
+    # StaticArrays' `reduce` is `foldl`:
+    results = _reduce(
+        a,
+        dims,
+        (init = (similar_type(a, Union{}, Size(0))(), init),),
+    ) do (ys, acc), x
+        y = rf(acc, x)
+        # Not using `push(ys, y)` here since we need to widen element type as
+        # we iterate.
+        (vcat(ys, SA[y]), y)
+    end
+    dims === (:) && return first(results)
+
+    ys = map(first, results)
+    # Now map over all indices of `a`.  Since `_map` needs at least
+    # one `StaticArray` to be passed, we pass `a` here, even though
+    # the values of `a` are not used.
+    data = _map(a, CartesianIndices(a)) do _, CI
+        D = _valof(dims)
+        I = Tuple(CI)
+        J = setindex(I, 1, D)
+        ys[J...][I[D]]
+    end
+    return similar_type(a, eltype(data))(data)
+end
+
+@inline Base.cumsum(a::StaticArray; kw...) = accumulate(Base.add_sum, a; kw...)
+@inline Base.cumprod(a::StaticArray; kw...) = accumulate(Base.mul_prod, a; kw...)
diff --git a/test/accumulate.jl b/test/accumulate.jl
new file mode 100644
index 00000000..97a23e7d
--- /dev/null
+++ b/test/accumulate.jl
@@ -0,0 +1,66 @@
+using StaticArrays, Test
+
+@testset "accumulate" begin
+    @testset "cumsum(::$label)" for (label, T) in [
+        # label, T
+        ("SVector", SVector),
+        ("MVector", MVector),
+        ("SizedVector", SizedVector),
+    ]
+        @testset "$label" for (label, a) in [
+            ("[1, 2, 3]", T{3}(SA[1, 2, 3])),
+            ("[]", T{0,Int}(())),
+        ]
+            @test cumsum(a) == cumsum(collect(a))
+            @test cumsum(a) isa similar_type(a)
+            @inferred cumsum(a)
+        end
+        @test eltype(cumsum(T{0,Int8}(()))) == eltype(cumsum(Int8[]))
+        @test eltype(cumsum(T{1,Int8}((1)))) == eltype(cumsum(Int8[1]))
+        @test eltype(cumsum(T{2,Int8}((1, 2)))) == eltype(cumsum(Int8[1, 2]))
+    end
+
+    @testset "cumsum(::$label; dims=2)" for (label, T) in [
+        # label, T
+        ("SMatrix", SMatrix),
+        ("MMatrix", MMatrix),
+        ("SizedMatrix", SizedMatrix),
+    ]
+        @testset "$label" for (label, a) in [
+            ("[1 2; 3 4; 5 6]", T{3,2}(SA[1 2; 3 4; 5 6])),
+            ("0 x 2 matrix", T{0,2,Float64}()),
+            ("2 x 0 matrix", T{2,0,Float64}()),
+        ]
+            @test cumsum(a; dims = 2) == cumsum(collect(a); dims = 2)
+            @test cumsum(a; dims = 2) isa similar_type(a)
+            v"1.1" <= VERSION < v"1.2" && continue
+            @inferred cumsum(a; dims = Val(2))
+        end
+    end
+
+    @testset "cumsum(a::SArray; dims=$i); ndims(a) = $d" for d in 1:4, i in 1:d
+        shape = Tuple(1:d)
+        a = similar_type(SArray, Int, Size(shape))(1:prod(shape))
+        @test cumsum(a; dims = i) == cumsum(collect(a); dims = i)
+        @test cumsum(a; dims = i) isa SArray
+        v"1.1" <= VERSION < v"1.2" && continue
+        @inferred cumsum(a; dims = Val(i))
+    end
+
+    @testset "cumprod" begin
+        a = SA[1, 2, 3]
+        @test cumprod(a)::SArray == cumprod(collect(a))
+        @inferred cumprod(a)
+
+        @test eltype(cumsum(SA{Int8}[])) == eltype(cumsum(Int8[]))
+        @test eltype(cumsum(SA{Int8}[1])) == eltype(cumsum(Int8[1]))
+        @test eltype(cumsum(SA{Int8}[1, 2])) == eltype(cumsum(Int8[1, 2]))
+    end
+
+    @testset "empty vector with init" begin
+        a = SA{Int}[]
+        right(_, x) = x
+        @test accumulate(right, a; init = Val(1)) === SA{Int}[]
+        @inferred accumulate(right, a; init = Val(1))
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index eee41e04..34b508fc 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -33,6 +33,7 @@ include("abstractarray.jl")
 include("indexing.jl")
 include("initializers.jl")
 Random.seed!(42); include("mapreduce.jl")
+Random.seed!(42); include("accumulate.jl")
 Random.seed!(42); include("arraymath.jl")
 include("broadcast.jl")
 include("linalg.jl")