diff --git a/src/adjtrans.jl b/src/adjtrans.jl
index 96db07d5..29f8d064 100644
--- a/src/adjtrans.jl
+++ b/src/adjtrans.jl
@@ -577,6 +577,8 @@ _vecadjoint(A::Base.ReshapedArray{<:Any,1,<:AdjointAbsVec}) = adjoint(parent(A))
 
 diagview(A::Transpose, k::Integer = 0) = _vectranspose(diagview(parent(A), -k))
 diagview(A::Adjoint, k::Integer = 0) = _vecadjoint(diagview(parent(A), -k))
+diag(A::Transpose, ::Val{k}) where {k} = _vectranspose(diag(parent(A), Val(-k)))
+diag(A::Adjoint, ::Val{k}) where {k} = _vecadjoint(diag(parent(A), Val(-k)))
 
 # triu and tril
 triu!(A::AdjOrTransAbsMat, k::Integer = 0) = wrapperop(A)(tril!(parent(A), -k))
diff --git a/src/bidiag.jl b/src/bidiag.jl
index cc5e6de7..0e36e7da 100644
--- a/src/bidiag.jl
+++ b/src/bidiag.jl
@@ -418,6 +418,9 @@ function triu!(M::Bidiagonal{T}, k::Integer=0) where T
     return M
 end
 
+diag(M::Bidiagonal, ::Val{0}) = M.dv
+diag(M::Bidiagonal, ::Val{1}) = M.uplo == 'U' ? M.ev : zero(M.ev)
+diag(M::Bidiagonal, ::Val{-1}) = M.uplo == 'L' ? M.ev : zero(M.ev)
 function diag(M::Bidiagonal, n::Integer=0)
     # every branch call similar(..., ::Int) to make sure the
     # same vector type is returned independent of n
diff --git a/src/dense.jl b/src/dense.jl
index f6f1accb..3f6ef78e 100644
--- a/src/dense.jl
+++ b/src/dense.jl
@@ -307,6 +307,22 @@ julia> diag(A,1)
 """
 diag(A::AbstractMatrix, k::Integer=0) = A[diagind(A, k, IndexStyle(A))]
 
+"""
+    diag(M, ::Val{k}) where {k}
+
+Return the `k`th diagonal of a matrix as a vector.
+For banded matrix types such as `Diagonal`, this may return the underlying
+band instead of making a copy if `k` lies within the bandwidth of the matrix.
+
+!!! note
+    The type of the result may vary depending on the values of `k`.
+"""
+function diag(A::AbstractMatrix, ::Val{k}) where {k}
+    # some types might have a specialized 1-arg `diag` method,
+    # and we may use this if possible
+    k == 0 ? diag(A) : diag(A, k)
+end
+
 """
     diagview(M, k::Integer=0)
 
diff --git a/src/diagonal.jl b/src/diagonal.jl
index e60fb009..ef2e7005 100644
--- a/src/diagonal.jl
+++ b/src/diagonal.jl
@@ -926,6 +926,7 @@ adjoint(D::Diagonal) = Diagonal(_vecadjoint(D.diag))
 permutedims(D::Diagonal) = D
 permutedims(D::Diagonal, perm) = (Base.checkdims_perm(axes(D), axes(D), perm); D)
 
+diag(D::Diagonal, ::Val{0}) = D.diag
 function diag(D::Diagonal, k::Integer=0)
     # every branch call similar(..., ::Int) to make sure the
     # same vector type is returned independent of k
diff --git a/src/hessenberg.jl b/src/hessenberg.jl
index 7aab1c77..b65f7be8 100644
--- a/src/hessenberg.jl
+++ b/src/hessenberg.jl
@@ -667,3 +667,6 @@ function logdet(F::Hessenberg)
     d,s = logabsdet(F)
     return d + log(s)
 end
+
+diag(A::UpperHessenberg) = diag(A.data)
+diag(A::UpperHessenberg, ::Val{k}) where {k} = k >= -1 ? diag(A.data, Val(k)) : diag(A, k)
diff --git a/src/triangular.jl b/src/triangular.jl
index d82ddd87..966e68e4 100644
--- a/src/triangular.jl
+++ b/src/triangular.jl
@@ -550,6 +550,12 @@ adjoint!(A::UnitUpperTriangular) = UnitLowerTriangular(copytri!(A.data, 'U' , tr
 
 diag(A::UpperOrLowerTriangular) = diag(A.data)
 diag(A::Union{UnitLowerTriangular, UnitUpperTriangular}) = fill(oneunit(eltype(A)), size(A,1))
+diag(A::UpperTriangular, ::Val{k}) where {k} = k >= 0 ?  diag(A.data, Val(k)) : diag(A, k)
+diag(A::LowerTriangular, ::Val{k}) where {k} = k <= 0 ? diag(A.data, Val(k)) : diag(A, k)
+diag(A::UnitUpperTriangular, ::Val{0}) = diag(A)
+diag(A::UnitLowerTriangular, ::Val{0}) = diag(A)
+diag(A::UnitUpperTriangular, ::Val{k}) where {k} = k > 0 ? diag(A.data, Val(k)) : diag(A, k)
+diag(A::UnitLowerTriangular, ::Val{k}) where {k} = k < 0 ? diag(A.data, Val(k)) : diag(A, k)
 
 # Unary operations
 -(A::LowerTriangular) = LowerTriangular(-A.data)
@@ -2994,8 +3000,8 @@ logdet(A::UnitUpperTriangular{T}) where {T} = zero(T)
 logdet(A::UnitLowerTriangular{T}) where {T} = zero(T)
 logabsdet(A::UnitUpperTriangular{T}) where {T} = zero(T), one(T)
 logabsdet(A::UnitLowerTriangular{T}) where {T} = zero(T), one(T)
-det(A::UpperTriangular) = prod(diag(A.data))
-det(A::LowerTriangular) = prod(diag(A.data))
+det(A::UpperTriangular) = prod(diag(A.data, Val(0)))
+det(A::LowerTriangular) = prod(diag(A.data, Val(0)))
 function logabsdet(A::Union{UpperTriangular{T},LowerTriangular{T}}) where T
     sgn = one(T)
     abs_det = zero(real(T))
diff --git a/src/tridiag.jl b/src/tridiag.jl
index 519be750..2426c096 100644
--- a/src/tridiag.jl
+++ b/src/tridiag.jl
@@ -111,7 +111,7 @@ function (::Type{SymTri})(A::AbstractMatrix) where {SymTri <: SymTridiagonal}
     checksquare(A)
     du = diag(A, 1)
     d  = diag(A)
-    if !(_issymmetric(A) || _checksymmetric(d, du, diag(A, -1)))
+    if !(_issymmetric(A) || _checksymmetric(d, du, diag(A, Val(-1))))
         throw(ArgumentError("matrix is not symmetric; cannot convert to SymTridiagonal"))
     end
     return SymTri(d, du)
@@ -195,6 +195,9 @@ _diagiter(M::SymTridiagonal) = (symmetric(x, :U) for x in M.dv)
 _eviter_transposed(M::SymTridiagonal{<:Number}) = _evview(M)
 _eviter_transposed(M::SymTridiagonal) = (transpose(x) for x in _evview(M))
 
+diag(M::SymTridiagonal{<:Number}, ::Val{0}) = M.dv
+diag(M::SymTridiagonal{<:Number}, ::Val{1}) = _evview(M)
+diag(M::SymTridiagonal{<:Number}, ::Val{-1}) = _evview(M)
 function diag(M::SymTridiagonal, n::Integer=0)
     # every branch call similar(..., ::Int) to make sure the
     # same vector type is returned independent of n
@@ -700,6 +703,9 @@ issymmetric(S::Tridiagonal) = all(issymmetric, S.d) && all(Iterators.map((x, y)
 
 \(A::Adjoint{<:Any,<:Tridiagonal}, B::Adjoint{<:Any,<:AbstractVecOrMat}) = copy(A) \ B
 
+diag(M::Tridiagonal, ::Val{0}) = M.d
+diag(M::Tridiagonal, ::Val{1}) = M.du
+diag(M::Tridiagonal, ::Val{-1}) = M.dl
 function diag(M::Tridiagonal, n::Integer=0)
     # every branch call similar(..., ::Int) to make sure the
     # same vector type is returned independent of n
diff --git a/test/adjtrans.jl b/test/adjtrans.jl
index 075f0acf..89d7cbd9 100644
--- a/test/adjtrans.jl
+++ b/test/adjtrans.jl
@@ -799,6 +799,25 @@ end
     end
 end
 
+@testset "diag with a Val index" begin
+    @testset "$(typeof(A))" for A in Any[rand(4, 4), rand(ComplexF64,4,4), fill([1 2; 3 4], 4, 4),
+            Diagonal(1:4), Bidiagonal(1:4, 1:3, :U),
+            Tridiagonal(1:3, 1:4, 1:3), SymTridiagonal(1:4, 1:3)]
+        @testset for (wrap_fn, wrap_T) in ((transpose,Transpose), (adjoint,Adjoint))
+            At = wrap_fn(A)
+            @test diag(At, 1) == diag(At, Val(1))
+            @test diag(At, 0) == diag(At, Val(0))
+            @test diag(At, -1) == diag(At, Val(-1))
+            if !(At isa wrap_T)
+                AT = wrap_T(A)
+                @test diag(At, Val(1)) == diag(AT, Val(1))
+                @test diag(At, Val(0)) == diag(AT, Val(0))
+                @test diag(At, Val(-1)) == diag(AT, Val(-1))
+            end
+        end
+    end
+end
+
 @testset "triu!/tril!" begin
     @testset for sz in ((4,4), (3,4), (4,3))
         A = rand(sz...)
diff --git a/test/bidiag.jl b/test/bidiag.jl
index 6c20a4b8..d9051511 100644
--- a/test/bidiag.jl
+++ b/test/bidiag.jl
@@ -1298,4 +1298,13 @@ end
     end
 end
 
+@testset "diag with a Val index" begin
+    B = Bidiagonal(1:4, 1:3, :U)
+    @test diag(B, Val(0)) === 1:4
+    @test diag(B, Val(1)) === 1:3
+    B = Bidiagonal(1:4, 1:3, :L)
+    @test diag(B, Val(0)) === 1:4
+    @test diag(B, Val(-1)) === 1:3
+end
+
 end # module TestBidiagonal
diff --git a/test/diagonal.jl b/test/diagonal.jl
index 712f426c..ce331483 100644
--- a/test/diagonal.jl
+++ b/test/diagonal.jl
@@ -1576,4 +1576,9 @@ end
     @test D == D2
 end
 
+@testset "diag with a Val index" begin
+    D = Diagonal(1:4)
+    @test diag(D, Val(0)) === 1:4
+end
+
 end # module TestDiagonal
diff --git a/test/hessenberg.jl b/test/hessenberg.jl
index 5b61c798..81e6655a 100644
--- a/test/hessenberg.jl
+++ b/test/hessenberg.jl
@@ -320,4 +320,14 @@ end
     @test U == U2
 end
 
+@testset "diag with a Val index" begin
+    H = UpperHessenberg(Tridiagonal(1:3, 1:4, 1:3))
+    @test diag(H, Val(0)) === 1:4
+    @test diag(H, Val(1)) === 1:3
+    @test diag(H, Val(-1)) === 1:3
+    @test diag(H, Val(0)) == diag(H) == diag(H, 0)
+    @test diag(H, Val(2)) == diag(H, 2)
+    @test diag(H, Val(-2)) == diag(H, -2)
+end
+
 end # module TestHessenberg
diff --git a/test/triangular.jl b/test/triangular.jl
index e823f698..61a39950 100644
--- a/test/triangular.jl
+++ b/test/triangular.jl
@@ -1110,4 +1110,24 @@ end
     end
 end
 
+@testset "diag with a Val index" begin
+    U = UpperTriangular(Tridiagonal(2:4, 1:4, 1:3))
+    @test diag(U, Val(0)) === 1:4
+    @test diag(U, Val(1)) === 1:3
+    @test diag(U, Val(-1)) == diag(U, -1) == zeros(3)
+    L = LowerTriangular(Tridiagonal(2:4, 1:4, 1:3))
+    @test diag(L, Val(0)) === 1:4
+    @test diag(L, Val(-1)) === 2:4
+    @test diag(L, Val(1)) == diag(L, 1) == zeros(3)
+
+    U = UnitUpperTriangular(Tridiagonal(2:4, 1:4, 1:3))
+    @test diag(U, Val(1)) === 1:3
+    @test diag(U, Val(0)) == diag(U, 0) == diag(U) == ones(4)
+    @test diag(U, Val(-1)) == diag(U, -1) == zeros(3)
+    L = UnitLowerTriangular(Tridiagonal(2:4, 1:4, 1:3))
+    @test diag(L, Val(-1)) === 2:4
+    @test diag(L, Val(0)) == diag(L, 0) == diag(L) == ones(4)
+    @test diag(L, Val(1)) == diag(L, 1) == zeros(3)
+end
+
 end # module TestTriangular
diff --git a/test/tridiag.jl b/test/tridiag.jl
index e955a37e..888557ac 100644
--- a/test/tridiag.jl
+++ b/test/tridiag.jl
@@ -1255,4 +1255,15 @@ end
     end
 end
 
+@testset "diag with a Val index" begin
+    T = Tridiagonal(2:4, 1:4, 1:3)
+    @test diag(T, Val(0)) === 1:4
+    @test diag(T, Val(1)) === 1:3
+    @test diag(T, Val(-1)) === 2:4
+    ST = SymTridiagonal(1:4, 1:3)
+    @test diag(ST, Val(0)) === 1:4
+    @test diag(ST, Val(1)) === 1:3
+    @test diag(ST, Val(-1)) === 1:3
+end
+
 end # module TestTridiagonal