diff --git a/src/rulesets/LinearAlgebra/dense.jl b/src/rulesets/LinearAlgebra/dense.jl
index a5edd6cd5..11f0c202b 100644
--- a/src/rulesets/LinearAlgebra/dense.jl
+++ b/src/rulesets/LinearAlgebra/dense.jl
@@ -394,3 +394,64 @@ function rrule(
     end
     return Ω, lyap_pullback
 end
+
+#####
+##### `kron`
+#####
+
+@static if VERSION ≥ v"1.9.0-DEV.1267"
+    function frule((_, Δx, Δy), ::typeof(kron), x::AbstractVecOrMat{<:Number}, y::AbstractVecOrMat{<:Number})
+        return kron(x, y), kron(Δx, y) + kron(x, Δy)
+    end
+
+    function rrule(::typeof(kron), x::AbstractVector{<:Number}, y::AbstractVector{<:Number})
+        project_x = ProjectTo(x)
+        project_y = ProjectTo(y)
+        function kron_pullback(z̄)
+            dz = reshape(unthunk(z̄), length(y), length(x))
+            x̄ = @thunk(project_x(conj.(dz' * y)))
+            ȳ = @thunk(project_y(dz * conj.(x)))
+            return NoTangent(), x̄, ȳ
+        end
+        return kron(x, y), kron_pullback
+    end
+
+    function rrule(::typeof(kron), x::AbstractMatrix{<:Number}, y::AbstractVector{<:Number})
+        project_x = ProjectTo(x)
+        project_y = ProjectTo(y)
+        function kron_pullback(z̄)
+            dz = reshape(unthunk(z̄), length(y), size(x)...)
+            x̄ = @thunk(project_x(dot.(Ref(y), eachslice(dz; dims = (2, 3)))))
+            ȳ = @thunk(project_y(dot.(Ref(x), eachslice(dz; dims = 1))))
+            return NoTangent(), x̄, ȳ
+        end
+        return kron(x, y), kron_pullback
+    end
+
+    function rrule(::typeof(kron), x::AbstractVector{<:Number}, y::AbstractMatrix{<:Number})
+        project_x = ProjectTo(x)
+        project_y = ProjectTo(y)
+        function kron_pullback(z̄)
+            dz = reshape(unthunk(z̄), size(y, 1), length(x), size(y, 2))
+            x̄ = @thunk(project_x(dot.(Ref(y), eachslice(dz; dims = 2))))
+            ȳ = @thunk(project_y(dot.(Ref(x), eachslice(dz; dims = (1, 3)))))
+            return NoTangent(), x̄, ȳ
+        end
+        return kron(x, y), kron_pullback
+    end
+
+    function rrule(::typeof(kron), x::AbstractMatrix{<:Number}, y::AbstractMatrix{<:Number})
+        project_x = ProjectTo(x)
+        project_y = ProjectTo(y)
+        function kron_pullback(z̄)
+            dz = reshape(unthunk(z̄), size(y, 1), size(x, 1), size(y, 2), size(x, 2))
+            x̄ = @thunk(project_x(_dot_collect.(Ref(y), eachslice(dz; dims = (2, 4)))))
+            ȳ = @thunk(project_y(_dot_collect.(Ref(x), eachslice(dz; dims = (1, 3)))))
+            return NoTangent(), x̄, ȳ
+        end
+        return kron(x, y), kron_pullback
+    end
+
+    _dot_collect(A::AbstractMatrix, B::SubArray) = dot(A, B)
+    _dot_collect(A::Diagonal, B::SubArray) = dot(A, collect(B))
+end
diff --git a/test/rulesets/LinearAlgebra/dense.jl b/test/rulesets/LinearAlgebra/dense.jl
index 5f5efa8d2..c145dfdbf 100644
--- a/test/rulesets/LinearAlgebra/dense.jl
+++ b/test/rulesets/LinearAlgebra/dense.jl
@@ -159,4 +159,26 @@
             test_rrule(lyap, A, C)
         end
     end
+    VERSION ≥ v"1.9.0" && @testset "kron" begin
+        @testset "AbstractVecOrMat{$T1}, AbstractVecOrMat{$T2}" for T1 in (Float64, ComplexF64), T2 in (Float64, ComplexF64)
+            @testset "frule" begin
+                test_frule(kron, randn(T1, 2), randn(T2, 3))
+                test_frule(kron, randn(T1, 2, 3), randn(T2, 5))
+                test_frule(kron, randn(T1, 2), randn(T2, 3, 5))
+                test_frule(kron, randn(T1, 2, 3), randn(T2, 5, 7))
+            end
+            @testset "rrule" begin
+                test_rrule(kron, randn(T1, 2), randn(T2, 3))
+
+                test_rrule(kron, Diagonal(randn(T1, 2)), randn(T2, 3))
+                test_rrule(kron, randn(T1, 2, 3), randn(T2, 5))
+
+                test_rrule(kron, randn(T1, 2), randn(T2, 3, 5))
+                test_rrule(kron, randn(T1, 2), Diagonal(randn(T2, 3)))
+
+                test_rrule(kron, randn(T1, 2, 3), randn(T2, 5, 7))
+                test_rrule(kron, Diagonal(randn(T1, 2)), Diagonal(randn(T2, 3)))
+            end
+        end
+    end
 end