diff --git a/Project.toml b/Project.toml
index 3187acf10..b99ae1b73 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "0.7.4"
+version = "0.7.5"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
diff --git a/src/rulesets/LinearAlgebra/structured.jl b/src/rulesets/LinearAlgebra/structured.jl
index f61a7a7fb..677039743 100644
--- a/src/rulesets/LinearAlgebra/structured.jl
+++ b/src/rulesets/LinearAlgebra/structured.jl
@@ -63,18 +63,84 @@ function rrule(::typeof(*), D::Diagonal{<:Real}, V::AbstractVector{<:Real})
 end
 
 #####
-##### `Symmetric`
+##### `Symmetric`/`Hermitian`
 #####
 
-function rrule(::Type{<:Symmetric}, A::AbstractMatrix)
-    function Symmetric_pullback(ȳ)
-        return (NO_FIELDS, @thunk(_symmetric_back(ȳ)))
+function frule((_, ΔA, _), T::Type{<:LinearAlgebra.HermOrSym}, A::AbstractMatrix, uplo)
+    return T(A, uplo), T(ΔA, uplo)
+end
+
+function rrule(T::Type{<:LinearAlgebra.HermOrSym}, A::AbstractMatrix, uplo)
+    Ω = T(A, uplo)
+    function HermOrSym_pullback(ΔΩ)
+        return (NO_FIELDS, @thunk(_symherm_back(T, ΔΩ, Ω.uplo)), DoesNotExist())
+    end
+    return Ω, HermOrSym_pullback
+end
+
+function frule((_, ΔA), TM::Type{<:Matrix}, A::LinearAlgebra.HermOrSym)
+    return TM(A), TM(_symherm_forward(A, ΔA))
+end
+function frule((_, ΔA), ::Type{Array}, A::LinearAlgebra.HermOrSym)
+    return Array(A), Array(_symherm_forward(A, ΔA))
+end
+
+function rrule(TM::Type{<:Matrix}, A::LinearAlgebra.HermOrSym)
+    function Matrix_pullback(ΔΩ)
+        TA = _symhermtype(A)
+        T∂A = TA{eltype(ΔΩ),typeof(ΔΩ)}
+        uplo = A.uplo
+        ∂A = T∂A(_symherm_back(A, ΔΩ, uplo), uplo)
+        return NO_FIELDS, ∂A
     end
-    return Symmetric(A), Symmetric_pullback
+    return TM(A), Matrix_pullback
 end
+rrule(::Type{Array}, A::LinearAlgebra.HermOrSym) = rrule(Matrix, A)
 
-_symmetric_back(ΔΩ) = UpperTriangular(ΔΩ) + transpose(LowerTriangular(ΔΩ)) - Diagonal(ΔΩ)
-_symmetric_back(ΔΩ::Union{Diagonal,UpperTriangular}) = ΔΩ
+# Get type (Symmetric or Hermitian) from type or matrix
+_symhermtype(::Type{<:Symmetric}) = Symmetric
+_symhermtype(::Type{<:Hermitian}) = Hermitian
+_symhermtype(A) = _symhermtype(typeof(A))
+
+# for Ω = Matrix(A::HermOrSym), push forward ΔA to get ∂Ω
+function _symherm_forward(A, ΔA)
+    TA = _symhermtype(A)
+    return if ΔA isa TA
+        ΔA
+    else
+        TA{eltype(ΔA),typeof(ΔA)}(ΔA, A.uplo)
+    end
+end
+
+# for Ω = HermOrSym(A, uplo), pull back ΔΩ to get ∂A
+_symherm_back(::Type{<:Symmetric}, ΔΩ, uplo) = _symmetric_back(ΔΩ, uplo)
+function _symherm_back(::Type{<:Hermitian}, ΔΩ::AbstractMatrix{<:Real}, uplo)
+    return _symmetric_back(ΔΩ, uplo)
+end
+_symherm_back(::Type{<:Hermitian}, ΔΩ, uplo) = _hermitian_back(ΔΩ, uplo)
+_symherm_back(Ω, ΔΩ, uplo) = _symherm_back(typeof(Ω), ΔΩ, uplo)
+
+function _symmetric_back(ΔΩ, uplo)
+    L, U, D = LowerTriangular(ΔΩ), UpperTriangular(ΔΩ), Diagonal(ΔΩ)
+    return uplo == 'U' ? U .+ transpose(L) - D : L .+ transpose(U) - D
+end
+_symmetric_back(ΔΩ::Diagonal, uplo) = ΔΩ
+_symmetric_back(ΔΩ::UpperTriangular, uplo) = Matrix(uplo == 'U' ? ΔΩ : transpose(ΔΩ))
+_symmetric_back(ΔΩ::LowerTriangular, uplo) = Matrix(uplo == 'U' ? transpose(ΔΩ) : ΔΩ)
+
+function _hermitian_back(ΔΩ, uplo)
+    L, U, rD = LowerTriangular(ΔΩ), UpperTriangular(ΔΩ), real.(Diagonal(ΔΩ))
+    return uplo == 'U' ? U .+ L' - rD : L .+ U' - rD
+end
+_hermitian_back(ΔΩ::Diagonal, uplo) = real.(ΔΩ)
+function _hermitian_back(ΔΩ::LinearAlgebra.AbstractTriangular, uplo)
+    ∂UL = ΔΩ .- Diagonal(_extract_imag.(diag(ΔΩ)))
+    return if istriu(ΔΩ)
+        return Matrix(uplo == 'U' ? ∂UL : ∂UL')
+    else
+        return Matrix(uplo == 'U' ? ∂UL' : ∂UL)
+    end
+end
 
 #####
 ##### `Adjoint`
diff --git a/src/rulesets/LinearAlgebra/utils.jl b/src/rulesets/LinearAlgebra/utils.jl
index caf661447..64dcdf7dc 100644
--- a/src/rulesets/LinearAlgebra/utils.jl
+++ b/src/rulesets/LinearAlgebra/utils.jl
@@ -34,3 +34,5 @@ function _add!(X::AbstractVecOrMat, Y::AbstractVecOrMat)
     return X
 end
 _add!(X, Y) = X + Y  # handles all `AbstractZero` overloads
+
+_extract_imag(x) = complex(0, imag(x))
diff --git a/test/rulesets/LinearAlgebra/structured.jl b/test/rulesets/LinearAlgebra/structured.jl
index 46fd77920..8ca1f0284 100644
--- a/test/rulesets/LinearAlgebra/structured.jl
+++ b/test/rulesets/LinearAlgebra/structured.jl
@@ -78,9 +78,47 @@
             end
         end
     end
-    @testset "Symmetric(::AbstractMatrix{$T})" for T in (Float64, ComplexF64)
+    @testset "$(SymHerm)(::AbstractMatrix{$T}, :$(uplo))" for
+        SymHerm in (Symmetric, Hermitian),
+        T in (Float64, ComplexF64),
+        uplo in (:U, :L)
+
+        N = 3
+        @testset "frule" begin
+            x = randn(T, N, N)
+            Δx = randn(T, N, N)
+            # can't use frule_test here because it doesn't yet ignore nothing tangents
+            Ω = SymHerm(x, uplo)
+            Ω_ad, ∂Ω_ad = frule((Zero(), Δx, Zero()), SymHerm, x, uplo)
+            @test Ω_ad == Ω
+            ∂Ω_fd = jvp(_fdm, z -> SymHerm(z, uplo), (x, Δx))
+            @test ∂Ω_ad ≈ ∂Ω_fd
+        end
+        @testset "rrule" begin
+            x = randn(T, N, N)
+            ∂x = randn(T, N, N)
+            ΔΩ = randn(T, N, N)
+            @testset "back(::$MT)" for MT in (Matrix, LowerTriangular, UpperTriangular)
+                rrule_test(SymHerm, MT(ΔΩ), (x, ∂x), (uplo, nothing))
+            end
+            @testset "back(::Diagonal)" begin
+                rrule_test(SymHerm, Diagonal(ΔΩ), (x, Diagonal(∂x)), (uplo, nothing))
+            end
+        end
+    end
+    @testset "$(f)(::$(SymHerm){$T}) with uplo=:$uplo" for f in (Matrix, Array),
+        SymHerm in (Symmetric, Hermitian),
+        T in (Float64, ComplexF64),
+        uplo in (:U, :L)
+
         N = 3
-        rrule_test(Symmetric, randn(T, N, N), (randn(T, N, N), randn(T, N, N)))
+        x = SymHerm(randn(T, N, N), uplo)
+        Δx = randn(T, N, N)
+        ∂x = SymHerm(randn(T, N, N), uplo)
+        ΔΩ = f(SymHerm(randn(T, N, N), uplo))
+        frule_test(f, (x, Δx))
+        frule_test(f, (x, SymHerm(Δx, uplo)))
+        rrule_test(f, ΔΩ, (x, ∂x))
     end
     @testset "$f" for f in (Adjoint, adjoint, Transpose, transpose)
         n = 5