diff --git a/src/hessenberg.jl b/src/hessenberg.jl
index d705c00e..a01a700b 100644
--- a/src/hessenberg.jl
+++ b/src/hessenberg.jl
@@ -59,6 +59,9 @@ size(H::UpperHessenberg) = size(H.data)
 axes(H::UpperHessenberg) = axes(H.data)
 parent(H::UpperHessenberg) = H.data
 
+upperhessenbergdata(H::UpperHessenberg) = H.data
+upperhessenbergdata(A) = A
+
 # similar behaves like UpperTriangular
 similar(H::UpperHessenberg, ::Type{T}) where {T} = UpperHessenberg(similar(H.data, T))
 similar(H::UpperHessenberg, ::Type{T}, dims::Dims{N}) where {T,N} = similar(H.data, T, dims)
diff --git a/src/structuredbroadcast.jl b/src/structuredbroadcast.jl
index 79a78dba..23c4d1d0 100644
--- a/src/structuredbroadcast.jl
+++ b/src/structuredbroadcast.jl
@@ -8,8 +8,12 @@ struct StructuredMatrixStyle{T} <: Broadcast.AbstractArrayStyle{2} end
 StructuredMatrixStyle{T}(::Val{2}) where {T} = StructuredMatrixStyle{T}()
 StructuredMatrixStyle{T}(::Val{N}) where {T,N} = Broadcast.DefaultArrayStyle{N}()
 
-const StructuredMatrix{T} = Union{Diagonal{T},Bidiagonal{T},SymTridiagonal{T},Tridiagonal{T},LowerTriangular{T},UnitLowerTriangular{T},UpperTriangular{T},UnitUpperTriangular{T}}
-for ST in (Diagonal,Bidiagonal,SymTridiagonal,Tridiagonal,LowerTriangular,UnitLowerTriangular,UpperTriangular,UnitUpperTriangular)
+const StructuredMatrix{T} = Union{Diagonal{T},Bidiagonal{T},SymTridiagonal{T},Tridiagonal{T},
+            LowerTriangular{T},UnitLowerTriangular{T},UpperTriangular{T},UnitUpperTriangular{T},
+            UpperHessenberg{T}}
+for ST in (Diagonal,Bidiagonal,SymTridiagonal,Tridiagonal,
+            LowerTriangular,UnitLowerTriangular,UpperTriangular,UnitUpperTriangular,
+            UpperHessenberg)
     @eval Broadcast.BroadcastStyle(::Type{<:$ST}) = $(StructuredMatrixStyle{ST}())
 end
 
@@ -27,28 +31,46 @@ Broadcast.BroadcastStyle(::StructuredMatrixStyle{Diagonal}, ::StructuredMatrixSt
     StructuredMatrixStyle{LowerTriangular}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{Diagonal}, ::StructuredMatrixStyle{<:Union{UpperTriangular,UnitUpperTriangular}}) =
     StructuredMatrixStyle{UpperTriangular}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{Diagonal}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
 
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{Bidiagonal}, ::StructuredMatrixStyle{Diagonal}) =
     StructuredMatrixStyle{Bidiagonal}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{Bidiagonal}, ::StructuredMatrixStyle{<:Union{Bidiagonal,SymTridiagonal,Tridiagonal}}) =
     StructuredMatrixStyle{Tridiagonal}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{Bidiagonal}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
+
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{SymTridiagonal}, ::StructuredMatrixStyle{<:Union{Diagonal,Bidiagonal,SymTridiagonal,Tridiagonal}}) =
     StructuredMatrixStyle{Tridiagonal}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{SymTridiagonal}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{Tridiagonal}, ::StructuredMatrixStyle{<:Union{Diagonal,Bidiagonal,SymTridiagonal,Tridiagonal}}) =
     StructuredMatrixStyle{Tridiagonal}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{Tridiagonal}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
 
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{LowerTriangular}, ::StructuredMatrixStyle{<:Union{Diagonal,LowerTriangular,UnitLowerTriangular}}) =
     StructuredMatrixStyle{LowerTriangular}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{UpperTriangular}, ::StructuredMatrixStyle{<:Union{Diagonal,UpperTriangular,UnitUpperTriangular}}) =
     StructuredMatrixStyle{UpperTriangular}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{UpperTriangular}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{UnitLowerTriangular}, ::StructuredMatrixStyle{<:Union{Diagonal,LowerTriangular,UnitLowerTriangular}}) =
     StructuredMatrixStyle{LowerTriangular}()
 Broadcast.BroadcastStyle(::StructuredMatrixStyle{UnitUpperTriangular}, ::StructuredMatrixStyle{<:Union{Diagonal,UpperTriangular,UnitUpperTriangular}}) =
     StructuredMatrixStyle{UpperTriangular}()
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{UnitUpperTriangular}, ::StructuredMatrixStyle{UpperHessenberg}) =
+    StructuredMatrixStyle{UpperHessenberg}()
+
+function Broadcast.BroadcastStyle(::StructuredMatrixStyle{UpperHessenberg},
+        ::StructuredMatrixStyle{<:Union{Diagonal,Bidiagonal,Tridiagonal,SymTridiagonal,UnitUpperTriangular,UpperTriangular}})
+    StructuredMatrixStyle{UpperHessenberg}()
+end
 
-Broadcast.BroadcastStyle(::StructuredMatrixStyle{<:Union{LowerTriangular,UnitLowerTriangular}}, ::StructuredMatrixStyle{<:Union{UpperTriangular,UnitUpperTriangular}}) =
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{<:Union{LowerTriangular,UnitLowerTriangular}}, ::StructuredMatrixStyle{<:Union{UpperTriangular,UnitUpperTriangular,UpperHessenberg}}) =
     StructuredMatrixStyle{Matrix}()
-Broadcast.BroadcastStyle(::StructuredMatrixStyle{<:Union{UpperTriangular,UnitUpperTriangular}}, ::StructuredMatrixStyle{<:Union{LowerTriangular,UnitLowerTriangular}}) =
+Broadcast.BroadcastStyle(::StructuredMatrixStyle{<:Union{UpperTriangular,UnitUpperTriangular,UpperHessenberg}}, ::StructuredMatrixStyle{<:Union{LowerTriangular,UnitLowerTriangular}}) =
     StructuredMatrixStyle{Matrix}()
 
 # Make sure that `StructuredMatrixStyle{Matrix}` doesn't ever end up falling
@@ -97,7 +119,7 @@ function structured_broadcast_alloc(bc, ::Type{Tridiagonal},
     Tridiagonal(Array{ElType}(undef, n1),Array{ElType}(undef, n),Array{ElType}(undef, n1))
 end
 function structured_broadcast_alloc(bc, ::Type{T}, ::Type{ElType},
-        sz::NTuple{2,Integer}) where {ElType,T<:UpperOrLowerTriangular}
+        sz::NTuple{2,Integer}) where {ElType,T<:Union{UpperOrLowerTriangular, UpperHessenberg}}
     T(Array{ElType}(undef, sz))
 end
 structured_broadcast_alloc(bc, ::Type{Matrix}, ::Type{ElType}, sz::NTuple{2,Integer}) where {ElType} =
@@ -278,6 +300,7 @@ function preprocess_broadcasted(::Type{T}, bc::Broadcasted) where {T}
 end
 _preprocess_broadcasted(::Type{LowerTriangular}, A) = lowertridata(A)
 _preprocess_broadcasted(::Type{UpperTriangular}, A) = uppertridata(A)
+_preprocess_broadcasted(::Type{UpperHessenberg}, A) = upperhessenbergdata(A)
 
 function copyto!(dest::LowerTriangular, bc::Broadcasted{<:StructuredMatrixStyle})
     isvalidstructbc(dest, bc) || return copyto!(dest, convert(Broadcasted{Nothing}, bc))
@@ -305,6 +328,19 @@ function copyto!(dest::UpperTriangular, bc::Broadcasted{<:StructuredMatrixStyle}
     return dest
 end
 
+function copyto!(dest::UpperHessenberg, bc::Broadcasted{<:StructuredMatrixStyle})
+    isvalidstructbc(dest, bc) || return copyto!(dest, convert(Broadcasted{Nothing}, bc))
+    axs = axes(dest)
+    axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
+    bc_unwrapped = preprocess_broadcasted(UpperHessenberg, bc)
+    for j in axs[2]
+        for i in 1:min(size(dest.data,1), j+1)
+            @inbounds dest.data[i,j] = bc_unwrapped[CartesianIndex(i, j)]
+        end
+    end
+    return dest
+end
+
 # We can also implement `map` and its promotion in terms of broadcast with a stricter dimension check
 function map(f, A::StructuredMatrix, Bs::StructuredMatrix...)
     sz = size(A)
diff --git a/test/structuredbroadcast.jl b/test/structuredbroadcast.jl
index 5e6a68d7..da9df54c 100644
--- a/test/structuredbroadcast.jl
+++ b/test/structuredbroadcast.jl
@@ -22,8 +22,9 @@ using .Main.SizedArrays
         S = SymTridiagonal(rand(N), rand(max(0,N-1)))
         U = UpperTriangular(rand(N,N))
         L = LowerTriangular(rand(N,N))
+        UH = UpperHessenberg(rand(N,N))
         M = Matrix(rand(N,N))
-        structuredarrays = (D, B, T, U, L, M, S)
+        structuredarrays = (D, B, T, U, L, M, S, UH)
         fstructuredarrays = map(Array, structuredarrays)
         @testset "$(nameof(typeof(X)))" for (X, fX) in zip(structuredarrays, fstructuredarrays)
             @test (Q = broadcast(sin, X); typeof(Q) == typeof(X) && Q == broadcast(sin, fX))
@@ -135,6 +136,7 @@ end
         T = Tridiagonal(rand(max(0,N-1)), rand(N), rand(max(0,N-1)))
         ◣ = LowerTriangular(rand(N,N))
         ◥ = UpperTriangular(rand(N,N))
+        UH = UpperHessenberg(rand(N,N))
         M = Matrix(rand(N,N))
 
         @test broadcast!(sin, copy(D), D)::Diagonal == sin.(D)::Diagonal
@@ -143,6 +145,7 @@ end
         @test broadcast!(sin, copy(T), T)::Tridiagonal == sin.(T)::Tridiagonal
         @test broadcast!(sin, copy(◣), ◣)::LowerTriangular == sin.(◣)::LowerTriangular
         @test broadcast!(sin, copy(◥), ◥)::UpperTriangular == sin.(◥)::UpperTriangular
+        @test broadcast!(sin, copy(UH), UH)::UpperHessenberg == sin.(UH)::UpperHessenberg
         @test broadcast!(sin, copy(M), M)::Matrix == sin.(M)::Matrix
         @test broadcast!(*, copy(D), D, A) == Diagonal(broadcast(*, D, A))
         @test broadcast!(*, copy(Bu), Bu, A) == Bidiagonal(broadcast(*, Bu, A), :U)
@@ -150,6 +153,7 @@ end
         @test broadcast!(*, copy(T), T, A) == Tridiagonal(broadcast(*, T, A))
         @test broadcast!(*, copy(◣), ◣, A) == LowerTriangular(broadcast(*, ◣, A))
         @test broadcast!(*, copy(◥), ◥, A) == UpperTriangular(broadcast(*, ◥, A))
+        @test broadcast!(*, copy(UH), UH, A) == UpperHessenberg(broadcast(*, UH, A))
         @test broadcast!(*, copy(M), M, A) == Matrix(broadcast(*, M, A))
 
         if N > 2
@@ -181,8 +185,9 @@ end
     S = SymTridiagonal(rand(N), rand(N - 1))
     U = UpperTriangular(rand(N,N))
     L = LowerTriangular(rand(N,N))
+    UH = UpperHessenberg(rand(N,N))
     M = Matrix(rand(N,N))
-    structuredarrays = (M, D, B, T, S, U, L)
+    structuredarrays = (M, D, B, T, S, U, L, UH)
     fstructuredarrays = map(Array, structuredarrays)
     for (X, fX) in zip(structuredarrays, fstructuredarrays)
         @test (Q = map(sin, X); typeof(Q) == typeof(X) && Q == map(sin, fX))
@@ -396,4 +401,11 @@ end
     end
 end
 
+@testset "Rectangular UpperHessenberg" begin
+    UH = UpperHessenberg(ones(4,3))
+    UH2 = UH .+ UH .- UH
+    @test UH2 == UH
+    @test UH2 isa UpperHessenberg
+end
+
 end