diff --git a/Project.toml b/Project.toml
index 8dc2bcf3..1c9d5d68 100644
--- a/Project.toml
+++ b/Project.toml
@@ -3,6 +3,7 @@ uuid = "f6369f11-7733-5829-9624-2563aa707210"
 version = "0.10.18"
 
 [deps]
+AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
 CommonSubexpressions = "bbf7d656-a473-5ed7-a52c-81e309532950"
 DiffResults = "163ba53b-c6d8-5494-b064-1a9d43ac40c5"
 DiffRules = "b552c78f-8df3-52c6-915a-8e097449b14b"
@@ -14,11 +15,13 @@ SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [compat]
+AbstractFFTs = "0.5, 1"
 Calculus = "0.2, 0.3, 0.4, 0.5"
 CommonSubexpressions = "0.3"
 DiffResults = "0.0.1, 0.0.2, 0.0.3, 0.0.4, 1.0.1"
 DiffRules = "0.0.4, 0.0.5, 0.0.6, 0.0.7, 0.0.8, 0.0.9, 0.0.10, 0.1, 1.0"
 DiffTests = "0.0.1, 0.1"
+FFTW = "1.2.4"
 NaNMath = "0.2.2, 0.3"
 SpecialFunctions = "0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.10, 1.0"
 StaticArrays = "0.8.3, 0.9, 0.10, 0.11, 0.12, 1.0"
@@ -27,9 +30,10 @@ julia = "1"
 [extras]
 Calculus = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
 DiffTests = "de460e47-3fe3-5279-bb4a-814414816d5d"
+FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
 InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Calculus", "DiffTests", "SparseArrays", "Test", "InteractiveUtils"]
+test = ["Calculus", "DiffTests", "FFTW", "SparseArrays", "Test", "InteractiveUtils"]
diff --git a/src/ForwardDiff.jl b/src/ForwardDiff.jl
index 51ce95a1..ca5e2b8c 100644
--- a/src/ForwardDiff.jl
+++ b/src/ForwardDiff.jl
@@ -21,6 +21,9 @@ include("gradient.jl")
 include("jacobian.jl")
 include("hessian.jl")
 
+import AbstractFFTs
+include("fft.jl")
+
 export DiffResults
 
 end # module
diff --git a/src/fft.jl b/src/fft.jl
new file mode 100644
index 00000000..09ba9d46
--- /dev/null
+++ b/src/fft.jl
@@ -0,0 +1,70 @@
+
+ForwardDiff.value(x::Complex{<:ForwardDiff.Dual}) =
+    Complex(x.re.value, x.im.value)
+
+ForwardDiff.partials(x::Complex{<:ForwardDiff.Dual}, n::Int) =
+    Complex(ForwardDiff.partials(x.re, n), ForwardDiff.partials(x.im, n))
+
+ForwardDiff.npartials(x::Complex{<:ForwardDiff.Dual{T,V,N}}) where {T,V,N} = N
+ForwardDiff.npartials(::Type{<:Complex{<:ForwardDiff.Dual{T,V,N}}}) where {T,V,N} = N
+
+# AbstractFFTs.complexfloat(x::AbstractArray{<:ForwardDiff.Dual}) = float.(x .+ 0im)
+AbstractFFTs.complexfloat(x::AbstractArray{<:ForwardDiff.Dual}) = AbstractFFTs.complexfloat.(x)
+AbstractFFTs.complexfloat(d::ForwardDiff.Dual{T,V,N}) where {T,V,N} = convert(ForwardDiff.Dual{T,float(V),N}, d) + 0im
+
+AbstractFFTs.realfloat(x::AbstractArray{<:ForwardDiff.Dual}) = AbstractFFTs.realfloat.(x)
+AbstractFFTs.realfloat(d::ForwardDiff.Dual{T,V,N}) where {T,V,N} = convert(ForwardDiff.Dual{T,float(V),N}, d)
+
+for plan in [:plan_fft, :plan_ifft, :plan_bfft]
+    @eval begin
+
+        AbstractFFTs.$plan(x::AbstractArray{<:ForwardDiff.Dual}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(ForwardDiff.value.(x) .+ 0im, region)
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:ForwardDiff.Dual}}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(ForwardDiff.value.(x), region)
+
+    end
+end
+
+# rfft only accepts real arrays
+AbstractFFTs.plan_rfft(x::AbstractArray{<:ForwardDiff.Dual}, region=1:ndims(x)) =
+    AbstractFFTs.plan_rfft(ForwardDiff.value.(x), region)
+
+for plan in [:plan_irfft, :plan_brfft]  # these take an extra argument, only when complex?
+    @eval begin
+
+        AbstractFFTs.$plan(x::AbstractArray{<:ForwardDiff.Dual}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(ForwardDiff.value.(x) .+ 0im, region)
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:ForwardDiff.Dual}}, d::Integer, region=1:ndims(x)) =
+            AbstractFFTs.$plan(ForwardDiff.value.(x), d, region)
+
+    end
+end
+
+for P in [:Plan, :ScaledPlan]  # need ScaledPlan to avoid ambiguities
+    @eval begin
+
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:ForwardDiff.Dual}) =
+            _apply_plan(p, x)
+
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Complex{<:ForwardDiff.Dual}}) =
+            _apply_plan(p, x)
+
+    end
+end
+
+function _apply_plan(p::AbstractFFTs.Plan, x::AbstractArray)
+    xtil = p * ForwardDiff.value.(x)
+    dxtils = ntuple(ForwardDiff.npartials(eltype(x))) do n
+        p * ForwardDiff.partials.(x, n)
+    end
+    map(xtil, dxtils...) do val, parts...
+        Complex(
+            ForwardDiff.Dual(real(val), map(real, parts)),
+            ForwardDiff.Dual(imag(val), map(imag, parts)),
+        )
+    end
+end
+
diff --git a/test/FFTTest.jl b/test/FFTTest.jl
new file mode 100644
index 00000000..aaeb1d8c
--- /dev/null
+++ b/test/FFTTest.jl
@@ -0,0 +1,26 @@
+module FFTTest
+
+using Test
+using ForwardDiff: Dual, valtype, value, partials
+using FFTW
+using AbstractFFTs: complexfloat, realfloat
+
+
+x1 = Dual.(1:4.0, 2:5, 3:6)
+
+@test value.(x1) == 1:4
+@test partials.(x1, 1) == 2:5
+
+@test complexfloat(x1)[1] === complexfloat(x1[1]) === Dual(1.0, 2.0, 3.0) + 0im
+@test realfloat(x1)[1] === realfloat(x1[1]) === Dual(1.0, 2.0, 3.0)
+
+@test fft(x1, 1)[1] isa Complex{<:Dual}
+
+@testset "$f" for f in [fft, ifft, rfft, bfft]
+    @test value.(f(x1)) == f(value.(x1))
+    @test partials.(f(x1), 1) == f(partials.(x1, 1))
+end
+
+
+
+end # module
diff --git a/test/runtests.jl b/test/runtests.jl
index 0b9b1d8b..87bebc04 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1,5 +1,9 @@
 using ForwardDiff
 
+println("Testing FFT...")
+t = @elapsed include("FFTTest.jl")
+println("done (took $t seconds).")
+
 println("Testing Partials...")
 t = @elapsed include("PartialsTest.jl")
 println("done (took $t seconds).")