diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index f83be7ce..620d6aee 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -16,7 +16,7 @@ jobs:
       fail-fast: false
       matrix:
         version:
-          - '1.0'
+          - '1.6'
           - '1'
           - 'nightly'
         os:
diff --git a/Project.toml b/Project.toml
index ea59c5fa..3eea1da7 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,8 +1,9 @@
 name = "ForwardDiff"
 uuid = "f6369f11-7733-5829-9624-2563aa707210"
-version = "0.10.33"
+version = "0.11"
 
 [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"
@@ -16,24 +17,27 @@ 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 = "1.4.0"
 DiffTests = "0.0.1, 0.1"
+FFTW = "1"
 LogExpFunctions = "0.3"
 NaNMath = "0.2.2, 0.3, 1"
 Preferences = "1"
 SpecialFunctions = "0.8, 0.9, 0.10, 1.0, 2"
 StaticArrays = "0.8.3, 0.9, 0.10, 0.11, 0.12, 1.0"
-julia = "1"
+julia = "1.6"
 
 [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 93d3b246..4b922d23 100644
--- a/src/ForwardDiff.jl
+++ b/src/ForwardDiff.jl
@@ -25,6 +25,9 @@ include("gradient.jl")
 include("jacobian.jl")
 include("hessian.jl")
 
+import AbstractFFTs
+include("fft.jl")
+
 export DiffResults
 
 end # module
diff --git a/src/dual.jl b/src/dual.jl
index 7e86e9b2..40c4e688 100644
--- a/src/dual.jl
+++ b/src/dual.jl
@@ -128,6 +128,8 @@ end
 @inline tagtype(::Type{V}) where {V} = Nothing
 @inline tagtype(::Dual{T,V,N}) where {T,V,N} = T
 @inline tagtype(::Type{Dual{T,V,N}}) where {T,V,N} = T
+@inline tagtype(::Complex{T}) where T = tagtype(T)
+@inline tagtype(::Type{Complex{T}}) where T = tagtype(T)
 
 ####################################
 # N-ary Operation Definition Tools #
diff --git a/src/fft.jl b/src/fft.jl
new file mode 100644
index 00000000..8df9bbb9
--- /dev/null
+++ b/src/fft.jl
@@ -0,0 +1,84 @@
+
+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
+
+# AbstractFFTs.complexfloat(x::AbstractArray{<:Dual}) = float.(x .+ 0im)
+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]
+    @eval begin
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(value.(x), region)
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:Dual}}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(value.(x), region)
+
+    end
+end
+
+# rfft only accepts real arrays
+AbstractFFTs.plan_rfft(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
+    AbstractFFTs.plan_rfft(value.(x), region)
+
+for plan in [:plan_irfft, :plan_brfft]  # these take an extra argument, only when complex?
+    @eval begin
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
+            AbstractFFTs.$plan(value.(x), region)
+
+        AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:Dual}}, d::Integer, region=1:ndims(x)) =
+            AbstractFFTs.$plan(value.(x), d, region)
+
+    end
+end
+
+# for f in (:dct, :idct)
+#     pf = Symbol("plan_", f)
+#     @eval begin
+#         AbstractFFTs.$f(x::AbstractArray{<:Dual}) = $pf(x) * x
+#         AbstractFFTs.$f(x::AbstractArray{<:Dual}, region) = $pf(x, region) * x
+#         AbstractFFTs.$pf(x::AbstractArray{<:Dual}, region; kws...) = $pf(value.(x), region; kws...)
+#         AbstractFFTs.$pf(x::AbstractArray{<:Complex}, region; kws...) = $pf(value.(x), region; kws...)
+#     end
+# end
+
+
+for P in [:Plan, :ScaledPlan]  # need ScaledPlan to avoid ambiguities
+    @eval begin
+
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Dual}) =
+            _apply_plan(p, x)
+
+        Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Complex{<:Dual}}) =
+            _apply_plan(p, x)
+
+    end
+end
+
+function _apply_plan(p::AbstractFFTs.Plan, x::AbstractArray)
+    xtil = p * value.(x)
+    dxtils = ntuple(npartials(eltype(x))) do n
+        p * partials.(x, n)
+    end
+    __apply_plan(tagtype(eltype(x)), xtil, dxtils)
+end
+
+function __apply_plan(T, xtil, dxtils)
+    map(xtil, dxtils...) do val, parts...
+        Complex(
+            Dual{T}(real(val), map(real, parts)),
+            Dual{T}(imag(val), map(imag, parts)),
+        )
+    end
+end
\ No newline at end of file
diff --git a/test/DualTest.jl b/test/DualTest.jl
index 0ec2beec..4dabdc9a 100644
--- a/test/DualTest.jl
+++ b/test/DualTest.jl
@@ -15,6 +15,7 @@ struct TestTag end
 struct OuterTestTag end
 
 samerng() = MersenneTwister(1)
+Random.seed!(132)
 
 # By lower-bounding the Int range at 2, we avoid cases where differentiating an
 # exponentiation of an Int value would cause a DomainError due to reducing the
diff --git a/test/FFTTest.jl b/test/FFTTest.jl
new file mode 100644
index 00000000..f3195eb1
--- /dev/null
+++ b/test/FFTTest.jl
@@ -0,0 +1,32 @@
+module FFTTest
+
+using Test
+using ForwardDiff: Dual, valtype, value, partials, derivative
+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
+
+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
+
+# c = x -> dct([x; 0; 0])[1]
+# @test derivative(c,0.1) ≈ 1
+end # module
diff --git a/test/runtests.jl b/test/runtests.jl
index a1ac67b3..6f62ce8f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -54,4 +54,5 @@ Random.seed!(SEED)
         end
     end
     println("##### Running all ForwardDiff tests took $(time() - t0) seconds.")
-end
\ No newline at end of file
+end
+