diff --git a/Project.toml b/Project.toml
index 1f12c7ba..ac9b4937 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,11 +1,13 @@
 name = "ForwardDiff"
 uuid = "f6369f11-7733-5829-9624-2563aa707210"
-version = "0.11-DEV"
+version = "0.11.0-DEV"
 
 [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"
+FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 LogExpFunctions = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
 NaNMath = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
@@ -16,17 +18,23 @@ SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [weakdeps]
+AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
+FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [extensions]
+ForwardDiffAbstractFFTsExt = "AbstractFFTs"
+ForwardDiffFFTWExt = "FFTW"
 ForwardDiffStaticArraysExt = "StaticArrays"
 
 [compat]
+AbstractFFTs = "1"
 Calculus = "0.5"
 CommonSubexpressions = "0.3"
 DiffResults = "1.1"
 DiffRules = "1.4"
 DiffTests = "0.1"
+FFTW = "1"
 LogExpFunctions = "0.3"
 NaNMath = "1"
 Preferences = "1"
@@ -35,12 +43,14 @@ StaticArrays = "1.5"
 julia = "1.6"
 
 [extras]
+AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
 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"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Calculus", "DiffTests", "SparseArrays", "StaticArrays", "Test", "InteractiveUtils"]
+test = ["AbstractFFTs", "Calculus", "DiffTests", "FFTW", "SparseArrays", "StaticArrays", "Test", "InteractiveUtils"]
diff --git a/ext/ForwardDiffAbstractFFTsExt.jl b/ext/ForwardDiffAbstractFFTsExt.jl
new file mode 100644
index 00000000..da0e035a
--- /dev/null
+++ b/ext/ForwardDiffAbstractFFTsExt.jl
@@ -0,0 +1,39 @@
+module ForwardDiffAbstractFFTsExt
+
+using ForwardDiff, AbstractFFTs
+
+import AbstractFFTs: plan_fft, plan_ifft, plan_bfft, plan_rfft, plan_brfft, plan_irfft, Plan
+using ForwardDiff: array2dual, dual2array, Dual
+import LinearAlgebra: mul!
+
+for P in (:Plan, :ScaledPlan)  # need ScaledPlan to avoid ambiguities
+    @eval begin
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{DT}) where DT<:Dual = array2dual(DT, p * dual2array(x))
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Complex{DT}}) where DT<:Dual = array2dual(DT, p * dual2array(x))
+    end
+end
+
+mul!(y::AbstractArray{<:Union{Dual,Complex{<:Dual}}}, p::Plan, x::AbstractArray{<:Union{Dual,Complex{<:Dual}}}) = copyto!(y, p*x)
+
+AbstractFFTs.complexfloat(x::AbstractArray{<:Dual}) = AbstractFFTs.complexfloat.(x)
+AbstractFFTs.complexfloat(d::Dual{T,V,N}) where {T,V,N} = convert(Dual{T,float(V),N}, d) + 0im
+
+AbstractFFTs.realfloat(x::AbstractArray{<:Dual}) = AbstractFFTs.realfloat.(x)
+AbstractFFTs.realfloat(d::Dual{T,V,N}) where {T,V,N} = convert(Dual{T,float(V),N}, d)
+
+for plan in (:plan_fft, :plan_ifft, :plan_bfft, :plan_rfft)
+    @eval begin
+        $plan(x::AbstractArray{<:Dual}, dims=1:ndims(x)) = $plan(dual2array(x), 1 .+ dims)
+        $plan(x::AbstractArray{<:Complex{<:Dual}}, dims=1:ndims(x)) = $plan(dual2array(x), 1 .+ dims)
+    end
+end
+
+for plan in (:plan_irfft, :plan_brfft)  # these take an extra argument, only when complex?
+    @eval begin
+        $plan(x::AbstractArray{<:Dual}, dims=1:ndims(x)) = $plan(dual2array(x), 1 .+ dims)
+        $plan(x::AbstractArray{<:Complex{<:Dual}}, d::Integer, dims=1:ndims(x)) = $plan(dual2array(x), d, 1 .+ dims)
+    end
+end
+
+
+end
\ No newline at end of file
diff --git a/ext/ForwardDiffFFTWExt.jl b/ext/ForwardDiffFFTWExt.jl
new file mode 100644
index 00000000..6e004962
--- /dev/null
+++ b/ext/ForwardDiffFFTWExt.jl
@@ -0,0 +1,14 @@
+module ForwardDiffFFTWExt
+
+using ForwardDiff: Dual, dual2array
+import FFTW: r2r, r2r!, plan_r2r
+
+
+plan_r2r(x::AbstractArray{<:Dual}, FLAG, dims=1:ndims(x)) = plan_r2r(dual2array(x), FLAG, 1 .+ dims)
+plan_r2r(x::AbstractArray{<:Complex{<:Dual}}, FLAG, dims=1:ndims(x)) = plan_r2r(dual2array(x), FLAG, 1 .+ dims)
+
+r2r(x::AbstractArray{<:Dual}, kinds, region...) = plan_r2r(x, kinds, region...) * x
+r2r(x::AbstractArray{<:Complex{<:Dual}}, kinds, region...) = plan_r2r(x, kinds, region...) * x
+
+
+end
\ No newline at end of file
diff --git a/src/ForwardDiff.jl b/src/ForwardDiff.jl
index fdfcd560..6e842af8 100644
--- a/src/ForwardDiff.jl
+++ b/src/ForwardDiff.jl
@@ -21,9 +21,12 @@ include("derivative.jl")
 include("gradient.jl")
 include("jacobian.jl")
 include("hessian.jl")
+include("complex.jl")
 
 if !isdefined(Base, :get_extension)
     include("../ext/ForwardDiffStaticArraysExt.jl")
+    include("../ext/ForwardDiffAbstractFFTsExt.jl")
+    include("../ext/ForwardDiffFFTWExt.jl")
 end
 
 export DiffResults
diff --git a/src/complex.jl b/src/complex.jl
new file mode 100644
index 00000000..0fabbc8d
--- /dev/null
+++ b/src/complex.jl
@@ -0,0 +1,14 @@
+@inline tagtype(::Complex{T}) where T = tagtype(T)
+@inline tagtype(::Type{Complex{T}}) where T = tagtype(T)
+
+dual2array(x::Array{<:Dual{Tag,T}}) where {Tag,T} = reinterpret(reshape, T, x)
+dual2array(x::Array{<:Complex{<:Dual{Tag, T}}}) where {Tag,T} = complex.(dual2array(real(x)), dual2array(imag(x)))
+array2dual(DT::Type{<:Dual}, x::Array{T}) where T = reinterpret(reshape, DT, real(x))
+array2dual(DT::Type{<:Dual}, x::Array{<:Complex{T}}) where T = complex.(array2dual(DT, real(x)), array2dual(DT, imag(x)))
+
+value(x::Complex{<:Dual}) = Complex(x.re.value, x.im.value)
+
+partials(x::Complex{<:Dual}, n::Int) = Complex(partials(x.re, n), partials(x.im, n))
+
+npartials(x::Complex{<:Dual{T,V,N}}) where {T,V,N} = N
+npartials(::Type{<:Complex{<:Dual{T,V,N}}}) where {T,V,N} = N
diff --git a/test/ComplexTest.jl b/test/ComplexTest.jl
new file mode 100644
index 00000000..604c13a5
--- /dev/null
+++ b/test/ComplexTest.jl
@@ -0,0 +1,11 @@
+module ComplexTest
+using ForwardDiff, Test
+using ForwardDiff: Dual, partials, value
+
+@testset "complex dual" begin
+    x = Dual(1., 2., 3.) + im*Dual(4.,5.,6.)
+    @test value(x) == 1 + 4im
+    @test partials(x,1) == 2 + 5im
+    @test partials(x,2) == 3 + 6im
+end
+end
\ No newline at end of file
diff --git a/test/FFTTest.jl b/test/FFTTest.jl
new file mode 100644
index 00000000..09bd6528
--- /dev/null
+++ b/test/FFTTest.jl
@@ -0,0 +1,82 @@
+module FFTTest
+
+using FFTW, LinearAlgebra, Test
+using ForwardDiff: Dual, valtype, value, partials, derivative
+using AbstractFFTs: complexfloat, realfloat
+
+@testset "fft and rfft" begin
+    x1 = Dual.(1:4.0, 2:5, 3:6)
+
+    @test value.(x1) == 1:4
+    @test partials.(x1, 1) == 2:5
+    @test partials.(x1, 2) == 3:6
+
+    @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))
+        @test partials.(f(x1), 2) == f(partials.(x1, 2))
+    end
+
+    @test ifft(fft(x1)) == x1
+    @test irfft(rfft(x1), length(x1)) ≈ x1
+    @test brfft(rfft(x1), length(x1)) ≈ 4x1
+
+    f = x -> real(fft([x; 0; 0])[1])
+    @test derivative(f,0.1) ≈ 1
+
+    r = x -> real(rfft([x; 0; 0])[1])
+    @test derivative(r,0.1) ≈ 1
+
+
+    n = 100
+    θ = range(0,2π; length=n+1)[1:end-1]
+    # emperical from Mathematical
+    @test derivative(ω -> fft(exp.(ω .* cos.(θ)))[1]/n, 1) ≈ 0.565159103992485
+
+    # c = x -> dct([x; 0; 0])[1]
+    # @test derivative(c,0.1) ≈ 1
+
+    @testset "matrix" begin
+        A = x1 * (1:10)'
+        @test value.(fft(A)) == fft(value.(A))
+        @test partials.(fft(A), 1) == fft(partials.(A, 1))
+        @test partials.(fft(A), 2) == fft(partials.(A, 2))
+
+        @test value.(fft(A, 1)) == fft(value.(A), 1)
+        @test partials.(fft(A, 1), 1) == fft(partials.(A, 1), 1)
+        @test partials.(fft(A, 1), 2) == fft(partials.(A, 2), 1)
+
+        @test value.(fft(A, 2)) == fft(value.(A), 2)
+        @test partials.(fft(A, 2), 1) == fft(partials.(A, 1), 2)
+        @test partials.(fft(A, 2), 2) == fft(partials.(A, 2), 2)
+    end
+
+    c1 = complex.(x1)
+    @test mul!(similar(c1), FFTW.plan_fft(x1), x1) == fft(x1)
+    @test mul!(similar(c1), FFTW.plan_fft(c1), c1) == fft(c1)
+end
+
+@testset "r2r" begin
+    x1 = Dual.(1:4.0, 2:5, 3:6)
+    t = FFTW.r2r(x1, FFTW.R2HC)
+
+    @test value.(t) == FFTW.r2r(value.(x1), FFTW.R2HC)
+    @test partials.(t, 1) == FFTW.r2r(partials.(x1, 1), FFTW.R2HC)
+    @test partials.(t, 2) == FFTW.r2r(partials.(x1, 2), FFTW.R2HC)
+
+    t = FFTW.r2r(x1 + 2im*x1, FFTW.R2HC)
+    @test value.(t) == FFTW.r2r(value.(x1 + 2im*x1), FFTW.R2HC)
+    @test partials.(t, 1) == FFTW.r2r(partials.(x1 + 2im*x1, 1), FFTW.R2HC)
+    @test partials.(t, 2) == FFTW.r2r(partials.(x1 + 2im*x1, 2), FFTW.R2HC)
+
+    f = ω -> FFTW.r2r([ω; zeros(9)], FFTW.R2HC)[1]
+    @test derivative(f, 0.1) ≡ 1.0
+
+    @test mul!(similar(x1), FFTW.plan_r2r(x1, FFTW.R2HC), x1) == FFTW.r2r(x1, FFTW.R2HC)
+end
+end # module
\ No newline at end of file
diff --git a/test/runtests.jl b/test/runtests.jl
index e440af65..14a044e6 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -51,5 +51,15 @@ Random.seed!(SEED)
         t = @elapsed include("AllocationsTest.jl")
         println("##### done (took $t seconds).")
     end
+    @testset "Complex" begin
+        println("##### Testing complex...")
+        t = @elapsed include("ComplexTest.jl")
+        println("##### done (took $t seconds).")
+    end
+    @testset "FFT" begin
+        println("##### Testing fft...")
+        t = @elapsed include("FFTTest.jl")
+        println("##### done (took $t seconds).")
+    end
     println("##### Running all ForwardDiff tests took $(time() - t0) seconds.")
 end