diff --git a/Project.toml b/Project.toml
index 0c208ae89..ab2c0f01f 100644
--- a/Project.toml
+++ b/Project.toml
@@ -3,6 +3,7 @@ uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
 version = "1.41.0"
 
 [deps]
+Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 Compat = "34da2185-b29b-5c13-b0c7-acf172513d20"
 Distributed = "8ba89e20-285c-5b6f-9357-94700520ee1b"
@@ -15,6 +16,7 @@ SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 
 [compat]
+Adapt = "3.4.0"
 ChainRulesCore = "1.15.3"
 ChainRulesTestUtils = "1.5"
 Compat = "3.42.0, 4"
@@ -28,7 +30,6 @@ StaticArrays = "1.2"
 julia = "1.6"
 
 [extras]
-Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
 ChainRulesTestUtils = "cdddcdb0-9152-4a09-a978-84456f9df70a"
 FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
 JLArrays = "27aeb0d3-9eb9-45fb-866b-73c2ecf80fcb"
@@ -38,4 +39,4 @@ StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Adapt", "ChainRulesTestUtils", "FiniteDifferences", "JLArrays", "JuliaInterpreter", "Random", "StaticArrays", "Test"]
+test = ["ChainRulesTestUtils", "FiniteDifferences", "JLArrays", "JuliaInterpreter", "Random", "StaticArrays", "Test"]
diff --git a/src/ChainRules.jl b/src/ChainRules.jl
index e323f7b6d..78ef7cd97 100644
--- a/src/ChainRules.jl
+++ b/src/ChainRules.jl
@@ -1,9 +1,11 @@
 module ChainRules
 
+using Adapt: adapt
 using Base.Broadcast: materialize, materialize!, broadcasted, Broadcasted, broadcastable
 using ChainRulesCore
 using Compat
 using Distributed
+using GPUArraysCore: AbstractGPUArray
 using IrrationalConstants: logtwo, logten
 using LinearAlgebra
 using LinearAlgebra.BLAS
diff --git a/src/rulesets/Base/base.jl b/src/rulesets/Base/base.jl
index 6bea6e06c..7b8df5b6c 100644
--- a/src/rulesets/Base/base.jl
+++ b/src/rulesets/Base/base.jl
@@ -241,3 +241,23 @@ function rrule(config::RuleConfig{>:HasReverseMode}, ::typeof(map), f::F, xs::Tu
     map_back(dy::AbstractZero) = (NoTangent(), NoTangent(), ntuple(Returns(NoTangent()), num_xs)...)
     return y, map_pullback
 end
+
+#####
+##### `task_local_storage`
+#####
+
+# Called by `@allowscalar` from GPUArrays
+
+ChainRules.@non_differentiable task_local_storage(key::Any)
+ChainRules.@non_differentiable task_local_storage(key::Any, value::Any)
+
+function rrule(config::RuleConfig{>:HasReverseMode}, ::typeof(task_local_storage), body::Function, key, value)
+    y, back = task_local_storage(key, value) do
+        rrule_via_ad(config, body)
+    end
+    function task_local_storage_pullback(dy)
+        dbody = only(back(dy))
+        return (NoTangent(), dbody, NoTangent(), NoTangent())
+    end
+    return y, task_local_storage_pullback
+end
diff --git a/src/rulesets/Base/indexing.jl b/src/rulesets/Base/indexing.jl
index a3d25ca82..d34c91c23 100644
--- a/src/rulesets/Base/indexing.jl
+++ b/src/rulesets/Base/indexing.jl
@@ -113,8 +113,6 @@ function ∇getindex!(dx::AbstractArray, x::AbstractArray, dy, inds::Integer...)
 end
 function ∇getindex!(dx::AbstractArray, x::AbstractArray, dy, inds...)
     view(dx, inds...) .+= dy
-    # For GPU arrays, `inds::Union{Integer, Base.Slice}...` is fine, but any other AbstractArray risks overwriting.
-    # Those should call `NNlib.scatter!`, alla https://github.com/FluxML/Zygote.jl/pull/1131
     return dx
 end
 
@@ -134,6 +132,25 @@ function rrule(::typeof(∇getindex), x, dy, inds...)
     return z, ∇getindex_pullback
 end
 
+# Indexing with repeated indices on a GPU will lead ∇getindex to have race conditions & wrong answers.
+# To avoid this, copy everything back to the CPU.
+# But don't do that for indices which are known to be unique, e.g. `A[1, 2:3, :]` the colon gives Base.Slice:
+
+function ∇getindex!(dx::AbstractGPUArray, x::AbstractArray, dy, inds::Integer...)
+    view(dx, inds...) .+= Ref(dy)
+    return dx
+end
+function ∇getindex!(dx::AbstractGPUArray, x::AbstractArray, dy, inds::Union{Integer, AbstractUnitRange, Base.Slice}...)
+    view(dx, inds...) .+= dy
+    return dx
+end
+function ∇getindex!(dx::AbstractGPUArray, x::AbstractArray, dy, inds...)
+    dx_cpu = adapt(Array, dx)
+    view(dx_cpu, adapt(Array, inds)...) .+= adapt(Array, dy)
+    copyto!(dx, dx_cpu)
+    return dx
+end
+
 #####
 ##### first, tail
 #####
diff --git a/test/rulesets/Base/indexing.jl b/test/rulesets/Base/indexing.jl
index 0d5b09398..c2b9203a4 100644
--- a/test/rulesets/Base/indexing.jl
+++ b/test/rulesets/Base/indexing.jl
@@ -143,6 +143,25 @@
         test_rrule(∇getindex, [rand(2) for _ in 1:3], rand(2), 3; check_inferred=false)
         test_rrule(∇getindex, [rand(2) for _ in 1:3], [rand(2), rand(2)], 1:2; check_inferred=false)
     end
+
+    @testset "GPU" begin
+        x_23_gpu = jl(rand(2, 3))
+    
+        # Scalar indexing, copied from:  @macroexpand @allowscalar A[i]
+        # Gives an error in Pkg.test, no idea why
+        # y1, bk1 = rrule(CFG, Base.task_local_storage, () -> x_23_gpu[1], :ScalarIndexing, ScalarAllowed)
+        # @test y1 == @allowscalar x_gpu[1]
+        # bk1(1.0)  # This is zero, because finite-differencing ignores the function
+        # ... but this works, and calls the rule:
+        # Zygote.gradient(x -> @allowscalar(x[1]), jl(rand(3)))[1]
+        
+        y2, bk2 = rrule(getindex, x_23_gpu, :, 2:3)  # fast path, just broadcast .+=
+        @test unthunk(bk2(jl(ones(2,2)))[2]) == jl([0 1 1; 0 1 1])
+
+        y3, bk3 = rrule(getindex, x_23_gpu, 1, [1,1,2])  # slow path, copy to CPU
+        @test_skip Array(y3) == Array(x_gpu)[1, [1,1,2]]  # error in Pkg.test, no idea why
+        @test unthunk(bk3(jl(ones(3)))[2]) == jl([2 1 0; 0 0 0])
+    end
 end
 
 @testset "first & tail" begin
@@ -178,6 +197,7 @@ end
 end
 
 @testset "unsafe_getindex" begin
+    # In real life this is called only on some AbstractRanges, but easier to test on Array:
     test_frule(Base.unsafe_getindex, collect(1:0.1:2), 3)
     test_rrule(Base.unsafe_getindex, collect(1:0.1:2), 3)
 end
diff --git a/test/runtests.jl b/test/runtests.jl
index 24c1d85b9..840c6de65 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -2,12 +2,15 @@ using Test, ChainRulesCore, ChainRulesTestUtils
 
 @nospecialize
 
+using Adapt
 using Base.Broadcast: broadcastable
 using ChainRules
 using ChainRulesCore
 using ChainRulesTestUtils
 using ChainRulesTestUtils: rand_tangent, _fdm
 using FiniteDifferences
+using GPUArraysCore
+using JLArrays
 using LinearAlgebra
 using LinearAlgebra.BLAS
 using LinearAlgebra: dot