diff --git a/Project.toml b/Project.toml
index f5c94dc2a..cd0497b6e 100644
--- a/Project.toml
+++ b/Project.toml
@@ -10,7 +10,7 @@ SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
 
 [compat]
 Cassette = "^0.2"
-FDM = "^0.5"
+FDM = "^0.6"
 julia = "^1.0"
 
 [extras]
diff --git a/src/rules/linalg/dense.jl b/src/rules/linalg/dense.jl
index 48a86de7c..6f86d25a3 100644
--- a/src/rules/linalg/dense.jl
+++ b/src/rules/linalg/dense.jl
@@ -1,3 +1,9 @@
+using LinearAlgebra: AbstractTriangular
+
+# Matrix wrapper types that we know are square and are thus potentially invertible. For
+# these we can use simpler definitions for `/` and `\`.
+const SquareMatrix{T} = Union{Diagonal{T},AbstractTriangular{T}}
+
 #####
 ##### `sum`
 #####
@@ -69,3 +75,74 @@ end
 frule(::typeof(tr), x) = (tr(x), Rule(Δx -> tr(extern(Δx))))
 
 rrule(::typeof(tr), x) = (tr(x), Rule(ΔΩ -> Diagonal(fill(ΔΩ, size(x, 1)))))
+
+#####
+##### `*`
+#####
+
+function rrule(::typeof(*), A::AbstractMatrix{<:Real}, B::AbstractMatrix{<:Real})
+    return A * B, (Rule(Ȳ -> Ȳ * B'), Rule(Ȳ -> A' * Ȳ))
+end
+
+#####
+##### `/`
+#####
+
+function rrule(::typeof(/), A::AbstractMatrix{<:Real}, B::T) where T<:SquareMatrix{<:Real}
+    Y = A / B
+    S = T.name.wrapper
+    ∂A = Rule(Ȳ -> Ȳ / B')
+    ∂B = Rule(Ȳ -> S(-Y' * (Ȳ / B')))
+    return Y, (∂A, ∂B)
+end
+
+function rrule(::typeof(/), A::AbstractVecOrMat{<:Real}, B::AbstractVecOrMat{<:Real})
+    Aᵀ, dA = rrule(adjoint, A)
+    Bᵀ, dB = rrule(adjoint, B)
+    Cᵀ, (dBᵀ, dAᵀ) = rrule(\, Bᵀ, Aᵀ)
+    C, dC = rrule(adjoint, Cᵀ)
+    ∂A = Rule(dA∘dAᵀ∘dC)
+    ∂B = Rule(dA∘dBᵀ∘dC)
+    return C, (∂A, ∂B)
+end
+
+#####
+##### `\`
+#####
+
+function rrule(::typeof(\), A::T, B::AbstractVecOrMat{<:Real}) where T<:SquareMatrix{<:Real}
+    Y = A \ B
+    S = T.name.wrapper
+    ∂A = Rule(Ȳ -> S(-(A' \ Ȳ) * Y'))
+    ∂B = Rule(Ȳ -> A' \ Ȳ)
+    return Y, (∂A, ∂B)
+end
+
+function rrule(::typeof(\), A::AbstractVecOrMat{<:Real}, B::AbstractVecOrMat{<:Real})
+    Y = A \ B
+    ∂A = Rule() do Ȳ
+        B̄ = A' \ Ȳ
+        Ā = -B̄ * Y'
+        _add!(Ā, (B - A * Y) * B̄' / A')
+        _add!(Ā, A' \ Y * (Ȳ' - B̄'A))
+        Ā
+    end
+    ∂B = Rule(Ȳ -> A' \ Ȳ)
+    return Y, (∂A, ∂B)
+end
+
+#####
+##### `norm`
+#####
+
+function rrule(::typeof(norm), A::AbstractArray{<:Real}, p::Real=2)
+    y = norm(A, p)
+    u = y^(1-p)
+    ∂A = Rule(ȳ -> ȳ .* u .* abs.(A).^p ./ A)
+    ∂p = Rule(ȳ -> ȳ * (u * sum(a->abs(a)^p * log(abs(a)), A) - y * log(y)) / p)
+    return y, (∂A, ∂p)
+end
+
+function rrule(::typeof(norm), x::Real, p::Real=2)
+    return norm(x, p), (Rule(ȳ -> ȳ * sign(x)), Rule(_ -> zero(x)))
+end
diff --git a/test/rules/linalg/dense.jl b/test/rules/linalg/dense.jl
index 097fd598d..0eb61cc6f 100644
--- a/test/rules/linalg/dense.jl
+++ b/test/rules/linalg/dense.jl
@@ -70,4 +70,68 @@ end
         frule_test(tr, (randn(rng, N, N), randn(rng, N, N)))
         rrule_test(tr, randn(rng), (randn(rng, N, N), randn(rng, N, N)))
     end
+    @testset "*" begin
+        rng = MersenneTwister(123456)
+        dims = [3,4,5]
+        for n in dims, m in dims, p in dims
+            n > 3 && n == m == p && continue  # don't need to test square case multiple times
+            A = randn(rng, m, n)
+            B = randn(rng, n, p)
+            Ȳ = randn(rng, m, p)
+            rrule_test(*, Ȳ, (A, randn(rng, m, n)), (B, randn(rng, n, p)))
+        end
+    end
+    @testset "$f" for f in [/, \]
+        rng = MersenneTwister(42)
+        for n in 3:5, m in 3:5
+            A = randn(rng, m, n)
+            B = randn(rng, m, n)
+            Ȳ = randn(rng, size(f(A, B)))
+            rrule_test(f, Ȳ, (A, randn(rng, m, n)), (B, randn(rng, m, n)))
+        end
+        # Vectors
+        x = randn(rng, 10)
+        y = randn(rng, 10)
+        ȳ = randn(rng, size(f(x, y))...)
+        rrule_test(f, ȳ, (x, randn(rng, 10)), (y, randn(rng, 10)))
+        if f == (/)
+            @testset "$T on the RHS" for T in (Diagonal, UpperTriangular, LowerTriangular)
+                RHS = T(randn(rng, T == Diagonal ? 10 : (10, 10)))
+                Y = randn(rng, 5, 10)
+                Ȳ = randn(rng, size(f(Y, RHS))...)
+                rrule_test(f, Ȳ, (Y, randn(rng, size(Y))), (RHS, randn(rng, size(RHS))))
+            end
+        else
+            @testset "$T on LHS" for T in (Diagonal, UpperTriangular, LowerTriangular)
+                LHS = T(randn(rng, T == Diagonal ? 10 : (10, 10)))
+                y = randn(rng, 10)
+                ȳ = randn(rng, size(f(LHS, y))...)
+                rrule_test(f, ȳ, (LHS, randn(rng, size(LHS))), (y, randn(rng, 10)))
+                Y = randn(rng, 10, 10)
+                Ȳ = randn(rng, 10, 10)
+                rrule_test(f, Ȳ, (LHS, randn(rng, size(LHS))), (Y, randn(rng, size(Y))))
+            end
+            @testset "Matrix $f Vector" begin
+                X = randn(rng, 10, 4)
+                y = randn(rng, 10)
+                ȳ = randn(rng, size(f(X, y))...)
+                rrule_test(f, ȳ, (X, randn(rng, size(X))), (y, randn(rng, 10)))
+            end
+            @testset "Vector $f Matrix" begin
+                x = randn(rng, 10)
+                Y = randn(rng, 10, 4)
+                ȳ = randn(rng, size(f(x, Y))...)
+                rrule_test(f, ȳ, (x, randn(rng, size(x))), (Y, randn(rng, size(Y))))
+            end
+        end
+    end
+    @testset "norm" begin
+        rng = MersenneTwister(3)
+        for dims in [(), (5,), (3, 2), (7, 3, 2)]
+            A = randn(rng, dims...)
+            p = randn(rng)
+            ȳ = randn(rng)
+            rrule_test(norm, ȳ, (A, randn(rng, dims...)), (p, randn(rng)))
+        end
+    end
 end