diff --git a/Project.toml b/Project.toml
index c51aa400..9d29ebe6 100644
--- a/Project.toml
+++ b/Project.toml
@@ -4,20 +4,24 @@ authors = ["Chris Rackauckas <accounts@chrisrackauckas.com>"]
 version = "1.8.0"
 
 [deps]
+ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 Requires = "ae029012-a4dd-5104-9daa-d747884805df"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
 [compat]
+ChainRulesCore = "0.9"
+FiniteDifferences = "0.11"
 Requires = "1.0"
 julia = "1"
 
 [extras]
+FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["ForwardDiff", "Test", "SafeTestsets", "Random"]
+test = ["FiniteDifferences", "ForwardDiff", "Test", "SafeTestsets", "Random"]
diff --git a/src/ExponentialUtilities.jl b/src/ExponentialUtilities.jl
index 7a584b15..4cddebf4 100644
--- a/src/ExponentialUtilities.jl
+++ b/src/ExponentialUtilities.jl
@@ -1,5 +1,5 @@
 module ExponentialUtilities
-using LinearAlgebra, SparseArrays, Printf, Requires
+using LinearAlgebra, SparseArrays, Printf, Requires, ChainRulesCore
 
 """
     @diagview(A,d) -> view of the `d`th diagonal of `A`.
@@ -20,6 +20,7 @@ include("krylov_phiv_adaptive.jl")
 include("kiops.jl")
 include("StegrWork.jl")
 include("krylov_phiv_error_estimate.jl")
+include("krylov_phiv_chainrules.jl")
 
 export phi, phi!, KrylovSubspace, arnoldi, arnoldi!, lanczos!, ExpvCache, PhivCache,
     expv, expv!, exp_generic, phiv, phiv!, kiops, expv_timestep, expv_timestep!, phiv_timestep, phiv_timestep!,
diff --git a/src/krylov_phiv_chainrules.jl b/src/krylov_phiv_chainrules.jl
new file mode 100644
index 00000000..f44e6f09
--- /dev/null
+++ b/src/krylov_phiv_chainrules.jl
@@ -0,0 +1,42 @@
+function ChainRulesCore.frule((_, Δt, ΔA, Δb), ::typeof(expv), t, A, b; kwargs...)
+    w = expv(t, A, b; kwargs...)
+    ∂w = similar(w)
+    mul!(∂w, A, w)
+    ∂w .*= Δt
+    if !isa(Δb, AbstractZero)
+        ∂w .+= expv(t, A, Δb; kwargs...)
+    end
+    # TODO: handle ΔA
+    ΔA isa AbstractZero || error("ΔA currently cannot be pushed forward")
+    return w, ∂w
+end
+
+function ChainRulesCore.rrule(::typeof(expv), t, A, b; kwargs...)
+    w = expv(t, A, b; kwargs...)
+    function expv_pullback(Δw)
+        ∂t = Thunk() do
+            t̄ = A isa AbstractMatrix ? conj(dot(Δw, A, w)) : dot(mul!(similar(w), A, w), Δw)
+            return t isa Real ? real(t̄) : t̄
+        end
+        # TODO: handle ∂A
+        ∂A = @thunk error("Adjoint wrt A not yet implemented")
+        ∂b = Thunk() do
+            # using similar is necessary to ensure type-stability
+            b̄ = similar(b)
+            _copyto!(b̄, expv(t', A', Δw; kwargs...))
+            return b̄
+        end
+        return (NO_FIELDS, ∂t, ∂A, ∂b)
+    end
+    expv_pullback(::Zero) = (NO_FIELDS, Zero(), Zero(), Zero())
+    return w, expv_pullback
+end
+
+function _copyto!(x, y)
+    if eltype(x) <: Real && !(eltype(y) <: Real)
+        x .= real.(y)
+    else
+        copyto!(x, y)
+    end
+    return x
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 4172affb..fa9e841c 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1,5 +1,6 @@
 using Test, LinearAlgebra, Random, SparseArrays, ExponentialUtilities
 using ExponentialUtilities: getH, getV, _exp!
+using ChainRulesCore, FiniteDifferences
 using ForwardDiff
 
 @testset "Exp" begin
@@ -228,3 +229,50 @@ struct OpnormFunctor end
         @test pv ≈ pv′ atol=1e-12
     end
 end
+
+@testset "expv chain rules" begin
+    n = 30
+    @testset "frule for T=$T" for T in (Float64, ComplexF64)
+        t = rand(T)
+        A = randn(T, n, n)
+        b = randn(T, n)
+        Δt = FiniteDifferences.rand_tangent(t)
+        Δb = FiniteDifferences.rand_tangent(b)
+
+        w = expv(t, A, b)
+        w_ad, ∂w_ad = frule((NO_FIELDS, Δt, Zero(), Δb), expv, t, A, b)
+        @test w_ad == w
+        ∂w_fd = jvp(central_fdm(5, 1), (t, b) -> expv(t, A, b), (t, Δt), (b, Δb))
+        @test ∂w_ad ≈ ∂w_fd
+
+        w_ad, ∂w_ad = frule((NO_FIELDS, Δt, Zero(), Zero()), expv, t, A, b)
+        @test w_ad == w
+        ∂w_fd = jvp(central_fdm(5, 1), t -> expv(t, A, b), (t, Δt))
+        @test ∂w_ad ≈ ∂w_fd
+
+        ΔA = FiniteDifferences.rand_tangent(A)
+        @test_throws ErrorException frule((NO_FIELDS, Δt, ΔA, Δb), expv, t, A, b)
+    end
+
+    @testset "rrule for T=$T" for T in (Float64, ComplexF64)
+        t = rand(T)
+        A = randn(T, n, n)
+        b = randn(T, n)
+        w = expv(t, A, b)
+        Δw = FiniteDifferences.rand_tangent(w)
+
+        w_ad, back = rrule(expv, t, A, b)
+        @test w_ad == w
+        ∂self, ∂t_ad, ∂A_ad, ∂b_ad = @inferred back(Δw)
+        @test ∂self === NO_FIELDS
+        @test @inferred(extern(∂t_ad)) isa typeof(t)
+        @test @inferred(extern(∂b_ad)) isa typeof(b)
+
+        ∂t_fd, ∂A_fd, ∂b_fd = j′vp(central_fdm(5, 1), expv, Δw, t, A, b)
+        @test extern(∂t_ad) ≈ ∂t_fd
+        @test extern(∂b_ad) ≈ ∂b_fd
+        @test_throws ErrorException unthunk(∂A_ad)
+
+        @test @inferred(back(Zero())) === (NO_FIELDS, Zero(), Zero(), Zero())
+    end
+end