diff --git a/Project.toml b/Project.toml
index 79b73556..77563c83 100644
--- a/Project.toml
+++ b/Project.toml
@@ -5,8 +5,11 @@ version = "0.3.9"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+Distances = "b4f34e82-e78d-54a5-968a-f98e89d6e8f7"
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
+Functors = "d9f16b24-f501-4c13-a1f2-28368ffc5196"
+GPUArrays = "0c68f7d7-f131-5f86-a1c3-88cf8149b2d7"
 KernelFunctions = "ec8451be-7e33-11e9-00cf-bbf324bd1392"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
diff --git a/examples/gpu-usage/Manifest.toml b/examples/gpu-usage/Manifest.toml
new file mode 100644
index 00000000..77120059
--- /dev/null
+++ b/examples/gpu-usage/Manifest.toml
@@ -0,0 +1,1088 @@
+# This file is machine-generated - editing it directly is not advised
+
+[[AbstractFFTs]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "485ee0867925449198280d4af84bdb46a2a404d0"
+uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
+version = "1.0.1"
+
+[[AbstractGPs]]
+deps = ["ChainRulesCore", "Distances", "Distributions", "FillArrays", "Functors", "GPUArrays", "KernelFunctions", "LinearAlgebra", "Random", "RecipesBase", "Reexport", "Statistics", "StatsBase", "Test"]
+path = "../.."
+uuid = "99985d1d-32ba-4be9-9821-2ec096f28918"
+version = "0.3.8"
+
+[[Adapt]]
+deps = ["LinearAlgebra"]
+git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7"
+uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
+version = "3.3.1"
+
+[[ArgTools]]
+uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
+
+[[ArrayInterface]]
+deps = ["IfElse", "LinearAlgebra", "Requires", "SparseArrays", "Static"]
+git-tree-sha1 = "cdb00a6fb50762255021e5571cf95df3e1797a51"
+uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
+version = "3.1.23"
+
+[[Artifacts]]
+uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"
+
+[[BFloat16s]]
+deps = ["LinearAlgebra", "Test"]
+git-tree-sha1 = "4af69e205efc343068dc8722b8dfec1ade89254a"
+uuid = "ab4f0b2a-ad5b-11e8-123f-65d77653426b"
+version = "0.1.0"
+
+[[Base64]]
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+
+[[Bzip2_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "c3598e525718abcc440f69cc6d5f60dda0a1b61e"
+uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
+version = "1.0.6+5"
+
+[[CEnum]]
+git-tree-sha1 = "215a9aa4a1f23fbd05b92769fdd62559488d70e9"
+uuid = "fa961155-64e5-5f13-b03f-caf6b980ea82"
+version = "0.4.1"
+
+[[CUDA]]
+deps = ["AbstractFFTs", "Adapt", "BFloat16s", "CEnum", "CompilerSupportLibraries_jll", "DataStructures", "ExprTools", "GPUArrays", "GPUCompiler", "LLVM", "LazyArtifacts", "Libdl", "LinearAlgebra", "Logging", "Printf", "Random", "Random123", "RandomNumbers", "Reexport", "Requires", "SparseArrays", "SpecialFunctions", "TimerOutputs"]
+git-tree-sha1 = "889889f1c13467406a126cd2789b4844487ddfc1"
+uuid = "052768ef-5323-5732-b1bb-66c8b64840ba"
+version = "3.3.5"
+
+[[Cairo_jll]]
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+git-tree-sha1 = "e2f47f6d8337369411569fd45ae5753ca10394c6"
+uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
+version = "1.16.0+6"
+
+[[ChainRulesCore]]
+deps = ["Compat", "LinearAlgebra", "SparseArrays"]
+git-tree-sha1 = "f53ca8d41e4753c41cdafa6ec5f7ce914b34be54"
+uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+version = "0.10.13"
+
+[[ColorSchemes]]
+deps = ["ColorTypes", "Colors", "FixedPointNumbers", "Random"]
+git-tree-sha1 = "9995eb3977fbf67b86d0a0a0508e83017ded03f2"
+uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
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+
+[[ColorTypes]]
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+git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597"
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+
+[[Colors]]
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+git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
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+
+[[CommonSubexpressions]]
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+uuid = "bbf7d656-a473-5ed7-a52c-81e309532950"
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+
+[[Compat]]
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+git-tree-sha1 = "344f143fa0ec67e47917848795ab19c6a455f32c"
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+
+[[CompilerSupportLibraries_jll]]
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+uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
+
+[[CompositionsBase]]
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+
+[[Contour]]
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+
+[[DataAPI]]
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+
+[[DataStructures]]
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+
+[[DataValueInterfaces]]
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+version = "1.0.0"
+
+[[Dates]]
+deps = ["Printf"]
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+
+[[DelimitedFiles]]
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+
+[[DiffResults]]
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+
+[[DiffRules]]
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+git-tree-sha1 = "85d2d9e2524da988bffaf2a381864e20d2dae08d"
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+
+[[Distances]]
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+git-tree-sha1 = "abe4ad222b26af3337262b8afb28fab8d215e9f8"
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+
+[[Distributed]]
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+uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"
+
+[[Distributions]]
+deps = ["FillArrays", "LinearAlgebra", "PDMats", "Printf", "QuadGK", "Random", "SparseArrays", "SpecialFunctions", "Statistics", "StatsBase", "StatsFuns"]
+git-tree-sha1 = "3889f646423ce91dd1055a76317e9a1d3a23fff1"
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+
+[[DocStringExtensions]]
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+[[Downloads]]
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+[[EarCut_jll]]
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+[[ExprTools]]
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+[[FFMPEG]]
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+[[Xorg_libxcb_jll]]
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+git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
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diff --git a/examples/gpu-usage/Project.toml b/examples/gpu-usage/Project.toml
new file mode 100644
index 00000000..91ec0ee7
--- /dev/null
+++ b/examples/gpu-usage/Project.toml
@@ -0,0 +1,16 @@
+[deps]
+AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
+CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
+Functors = "d9f16b24-f501-4c13-a1f2-28368ffc5196"
+Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306"
+LogExpFunctions = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
+Optim = "429524aa-4258-5aef-a3af-852621145aeb"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+
+[compat]
+AbstractGPs = "0.3"
+CUDA = "3"
+Functors = "0.2"
+Literate = "2.5"
+Optim = "1"
+Plots = "1"
diff --git a/examples/gpu-usage/script.jl b/examples/gpu-usage/script.jl
new file mode 100644
index 00000000..43a89e46
--- /dev/null
+++ b/examples/gpu-usage/script.jl
@@ -0,0 +1,83 @@
+# # A Minimal Working Example of AbstractGPs on CUDA
+#
+# ## Setup
+
+using AbstractGPs
+using CUDA
+using Optim
+using Functors
+using LogExpFunctions
+
+using Plots
+default(; legend=:outertopright, size=(700, 400))
+
+using Random
+Random.seed!(4321)
+#md nothing #hide
+
+# First, a function to create the kernel.
+# N.B. we can't use a ScaledKernel because of the hardcoded Vector -
+# https://github.com/JuliaGaussianProcesses/KernelFunctions.jl/issues/299
+
+make_kernel(ℓ_inv) = Matern52Kernel() ∘ ScaleTransform(ℓ_inv)
+#md nothing #hide
+
+# Next, sample some data from a GP
+
+lik_noise = 0.01
+f = GP(make_kernel(1.0))
+x = rand(20)
+fx = f(x, lik_noise)
+y = rand(fx)
+#md nothing #hide
+
+# Move everything to the GPU and compute the covariance
+
+gpu(f) = fmap(cu, f)
+
+x, y = gpu(x), gpu(y)
+fx = f(x)
+cov(fx) isa CuArray
+
+# Create an objective function and optimise the kernel parameters and inducing
+# points
+
+jitter = 1e-4
+function objective_function(x, y)
+    function negative_elbo(params)
+        kernel = make_kernel(params[1])
+        f = GP(kernel)
+        fx = f(x, lik_noise)
+        z = logistic.(params[2:end])
+        fz = f(z, jitter)  # "observing" the latent process with some (small) amount of jitter improves numerical stability
+        return -elbo(fx, y, fz)
+    end
+    return negative_elbo
+end
+
+x0 = cu(rand(6) * 2)
+# TODO: this doesn't work with CUDA.allowscalar(false) because of Optim
+opt = optimize(objective_function(x, y), x0, LBFGS())
+#md nothing #hide
+
+# Construct the approximate posterior using the optimised parameters
+
+opt_kernel = make_kernel(opt.minimizer[1])
+opt_f = GP(opt_kernel)
+opt_fx = opt_f(x, lik_noise)
+ap = approx_posterior(VFE(), opt_fx, y, opt_f(logistic.(opt.minimizer[2:end]), jitter))
+
+# Plot the optimised posterior (requires some casting to and from CuArrays)
+
+scatter(
+    Array(x),
+    Array(y);
+    xlim=(0, 1),
+    xlabel="x",
+    ylabel="y",
+    title="posterior (VI with sparse grid)",
+    label="Train Data",
+)
+x_plot = cu(collect(0:0.001:1))
+plot!(x_plot, Array(mean(ap(x_plot))); label=false)
+vline!(logistic.(opt.minimizer[3:end]); label="Pseudo-points")
diff --git a/src/AbstractGPs.jl b/src/AbstractGPs.jl
index e2aa4c96..e0a0e206 100644
--- a/src/AbstractGPs.jl
+++ b/src/AbstractGPs.jl
@@ -10,6 +10,9 @@ using Random
 using Statistics
 using StatsBase
 using RecipesBase
+using Functors
+using Distances
+using GPUArrays
 
 using KernelFunctions: ColVecs, RowVecs
 
@@ -62,6 +65,9 @@ include(joinpath("util", "plotting.jl"))
 # Testing utilities.
 include(joinpath("util", "test_util.jl"))
 
+# GPU compatibility
+include(joinpath("util", "gpu.jl"))
+
 # Deprecations.
 include("deprecations.jl")
 
diff --git a/src/finite_gp_projection.jl b/src/finite_gp_projection.jl
index 0c1c6bef..d595a876 100644
--- a/src/finite_gp_projection.jl
+++ b/src/finite_gp_projection.jl
@@ -10,6 +10,8 @@ struct FiniteGP{Tf<:AbstractGP,Tx<:AbstractVector,TΣ} <: AbstractMvNormal
     Σy::TΣ
 end
 
+@functor FiniteGP
+
 function FiniteGP(f::AbstractGP, x::AbstractVector, σ²::AbstractVector{<:Real})
     return FiniteGP(f, x, Diagonal(σ²))
 end
@@ -17,7 +19,7 @@ end
 const default_σ² = 1e-18
 
 function FiniteGP(f::AbstractGP, x::AbstractVector, σ²::Real=default_σ²)
-    return FiniteGP(f, x, Fill(σ², length(x)))
+    return FiniteGP(f, x, _fill_like(x, σ²))
 end
 
 function FiniteGP(
diff --git a/src/mean_function.jl b/src/mean_function.jl
index 3b16d72f..6eacec36 100644
--- a/src/mean_function.jl
+++ b/src/mean_function.jl
@@ -10,7 +10,7 @@ struct ZeroMean{T<:Real} <: MeanFunction end
 """
 This is an AbstractGPs-internal workaround for AD issues; ideally we would just extend Base.map
 """
-_map_meanfunction(::ZeroMean{T}, x::AbstractVector) where {T} = zeros(T, length(x))
+_map_meanfunction(::ZeroMean{T}, x::AbstractVector) where {T} = Zeros{T}(length(x))
 
 function ChainRulesCore.rrule(::typeof(_map_meanfunction), m::ZeroMean, x::AbstractVector)
     map_ZeroMean_pullback(Δ) = (NoTangent(), NoTangent(), ZeroTangent())
@@ -28,7 +28,7 @@ struct ConstMean{T<:Real} <: MeanFunction
     c::T
 end
 
-_map_meanfunction(m::ConstMean, x::AbstractVector) = fill(m.c, length(x))
+_map_meanfunction(m::ConstMean, x::AbstractVector) = Fill(m.c, length(x))
 
 """
     CustomMean{Tf} <: MeanFunction
diff --git a/src/util/gpu.jl b/src/util/gpu.jl
new file mode 100644
index 00000000..836497d2
--- /dev/null
+++ b/src/util/gpu.jl
@@ -0,0 +1,60 @@
+# TODO: these should be in KernelFunctions
+function KernelFunctions.pairwise(
+    ::SqEuclidean, x::AbstractGPUArray{<:Real,2}, y::AbstractGPUArray{<:Real,2}, kwargs...
+)
+    return sum(abs2, x; dims=1)' .+ sum(abs2, y; dims=1) .- 2x'y
+end
+
+function KernelFunctions.pairwise(d::SqEuclidean, x::AbstractGPUArray{<:Real,2}, kwargs...)
+    return KernelFunctions.pairwise(d, x, x)
+end
+
+function KernelFunctions.pairwise(
+    ::Euclidean, x::AbstractGPUArray{<:Real,2}, y::AbstractGPUArray{<:Real,2}; kwargs...
+)
+    return .√KernelFunctions.pairwise(SqEuclidean(), x, y)
+end
+function KernelFunctions.pairwise(d::Euclidean, x::AbstractGPUArray{<:Real,2}; kwargs...)
+    return KernelFunctions.pairwise(d, x, x)
+end
+
+function KernelFunctions.pairwise(
+    ::Euclidean, x::AbstractGPUArray, y::AbstractGPUArray; kwargs...
+)
+    return .√KernelFunctions.pairwise(SqEuclidean(), reshape(x, 1, :), reshape(y, 1, :))
+end
+function KernelFunctions.pairwise(d::Euclidean, x::AbstractGPUArray; kwargs...)
+    return KernelFunctions.pairwise(d, x, x)
+end
+
+#TODO: overload colwise for var(fx)
+
+# Temporarily needed until var(fx) is working
+function tr_Cf_invΣy(f::FiniteGP, Σy::Diagonal{<:Any,<:AbstractGPUArray}, chol_Σy::Cholesky)
+    return sum(diag(cov(f.f, f.x)) ./ diag(Σy))
+end
+
+# Fixes a bug with cholesky of Diagonal CuArrays
+function _cholesky(X::Diagonal{<:Real,T}) where {T<:AbstractGPUArray}
+    # Bit of a hack - should maybe specialise to CuArray?
+    return cholesky(T.name.wrapper(X))
+end
+
+_fill_like(x::AbstractVector, σ²) = similar(x) .= σ²
+_fill_like(x::Union{ColVecs,RowVecs}, σ²) = similar(x[1], length(x)) .= σ²
+
+# # If we want to use FillArrays for fx.Σy, something like the following is needed in cov(fx):
+
+# _add_broadcasted(A, B) = A .+ B
+
+# # TODO: this is potentially much harder for var(fx)
+# function _add_broadcasted(A::AbstractGPUArray, B::Diagonal{T, <:FillArrays.AbstractFill}) where T
+#     return Base.materialize(Base.broadcasted(Base.BroadcastStyle(typeof(A)), +, A, B))
+# end
+# _add_broadcasted(B::Diagonal{T, <:FillArrays.AbstractFill}, A::AbstractGPUArray) where T = _add_broadcasted(A, B)
+
+# # A more general approach?
+# const WrappedFillArray{T,N} = Adapt.WrappedArray{T, N, FillArrays.AbstractFill,FillArrays.AbstractFill{T,N}}
+# function _add_broadcasted(A::AbstractGPUArray, B::WrappedFillArray) where T
+#     Base.materialize(Base.broadcasted(Base.BroadcastStyle(typeof(A)), +, A, B))
+# end