diff --git a/src/rulesets/Base/array.jl b/src/rulesets/Base/array.jl
index 63ce70ae0..2cd03ce9e 100644
--- a/src/rulesets/Base/array.jl
+++ b/src/rulesets/Base/array.jl
@@ -65,9 +65,32 @@ function rrule(::typeof(reshape), A::AbstractArray, dims::Union{Colon,Int}...)
     return reshape(A, dims...), reshape_pullback
 end
 
+#####
+##### `permutedims`
+#####
+
+function rrule(::typeof(permutedims), x::AbstractVector)
+    project = ProjectTo(x)
+    permutedims_pullback_1(dy) = (NoTangent(), project(permutedims(unthunk(dy))))
+    return permutedims(x), permutedims_pullback_1
+end
+
+function rrule(::typeof(permutedims), x::AbstractArray, perm)
+    pr = ProjectTo(x)  # projection restores e.g. transpose([1,2,3])
+    permutedims_back_2(dy) = (NoTangent(), pr(permutedims(unthunk(dy), invperm(perm))), NoTangent())
+    return permutedims(x, perm), permutedims_back_2
+end
+
+function rrule(::typeof(PermutedDimsArray), x::AbstractArray, perm)
+    pr = ProjectTo(x)
+    permutedims_back_3(dy) = (NoTangent(), pr(permutedims(unthunk(dy), invperm(perm))), NoTangent())
+    return PermutedDimsArray(x, perm), permutedims_back_3
+end
+
 #####
 ##### `repeat`
 #####
+
 function rrule(::typeof(repeat), xs::AbstractArray; inner=ntuple(Returns(1), ndims(xs)), outer=ntuple(Returns(1), ndims(xs)))
 
     project_Xs = ProjectTo(xs)
diff --git a/test/rulesets/Base/array.jl b/test/rulesets/Base/array.jl
index ff3e93931..96be78198 100644
--- a/test/rulesets/Base/array.jl
+++ b/test/rulesets/Base/array.jl
@@ -42,6 +42,22 @@ end
     test_rrule(reshape, rand(4, 5), 2, :)
 end
 
+@testset "permutedims + PermutedDimsArray" begin
+    test_rrule(permutedims, rand(5))
+
+    test_rrule(permutedims, rand(3, 4), (2, 1))
+    test_rrule(permutedims, Diagonal(rand(5)), (2, 1))
+    # Note BTW that permutedims(Diagonal(rand(5))) does not use the rule at all
+
+    @test invperm((3, 1, 2)) != (3, 1, 2)
+    test_rrule(permutedims, rand(3, 4, 5), (3, 1, 2); check_inferred=VERSION>=v"1.1")
+
+    @test_skip test_rrule(PermutedDimsArray, rand(3, 4, 5), (3, 1, 2))
+    x = rand(2, 3, 4)
+    dy = rand(4, 2, 3)
+    @test rrule(permutedims, x, (3, 1, 2))[2](dy)[2] == rrule(PermutedDimsArray, x, (3, 1, 2))[2](dy)[2]
+end
+
 @testset "repeat" begin
 
     test_rrule(repeat, rand(4, ))