diff --git a/lib/cublas/linalg.jl b/lib/cublas/linalg.jl
index c8b205a2a5..a959fad268 100644
--- a/lib/cublas/linalg.jl
+++ b/lib/cublas/linalg.jl
@@ -37,6 +37,16 @@ end
 LinearAlgebra.norm(x::DenseCuArray{<:CublasFloat}) = nrm2(x)
 LinearAlgebra.BLAS.asum(x::StridedCuArray{<:CublasFloat}) = asum(length(x), x)
 
+function LinearAlgebra.norm(x::DenseCuArray{<:CublasFloat}, p::Integer)
+    if p==1
+        return CUBLAS.asum(length(x),x)
+    elseif p==2
+        return LinearAlgebra.norm(x)
+    else
+        return LinearAlgebra.tr(LinearAlgebra.Diagonal(abs.(x))^p)^(1/p)
+    end
+end
+
 function LinearAlgebra.axpy!(alpha::Number, x::StridedCuArray{T}, y::StridedCuArray{T}) where T<:CublasFloat
     length(x)==length(y) || throw(DimensionMismatch("axpy arguments have lengths $(length(x)) and $(length(y))"))
     axpy!(length(x), alpha, x, y)
diff --git a/test/cublas.jl b/test/cublas.jl
index 4319abb6bc..3e00e64d80 100644
--- a/test/cublas.jl
+++ b/test/cublas.jl
@@ -1,5 +1,6 @@
 using CUDA.CUBLAS
 using CUDA.CUBLAS: band, bandex
+using GPUArrays
 
 using LinearAlgebra
 
@@ -84,6 +85,14 @@ end
             mul!(y, f(A), x, Ts(1), Ts(2))
             @test Array(dy) ≈ y
         end
+
+        @testset "norm" begin
+            x, y, z = CUDA.rand(elty, 1), CUDA.rand(elty, 2), CUDA.rand(elty, m)
+            @disallowscalar @test typeof(norm(x,1)) <: Real
+            @disallowscalar @test typeof(norm(y,2)) <: Real
+            @disallowscalar @test typeof(norm(z,m)) <: Real
+        end
+
         @testset "banded methods" begin
             # bands
             ku = 2