diff --git a/lib/cusparse/conversions.jl b/lib/cusparse/conversions.jl
index f21dd8ace6..c016f9e83c 100644
--- a/lib/cusparse/conversions.jl
+++ b/lib/cusparse/conversions.jl
@@ -1,6 +1,45 @@
 # conversion routines between different sparse and dense storage formats
 
-SparseArrays.sparse(::DenseCuArray, args...) = error("CUSPARSE supports multiple sparse formats, use specific constructors instead (e.g. CuSparseMatrixCSC)")
+"""
+    sparse(x::DenseCuMatrix; fmt=:csc)
+    sparse(I::CuVector, J::CuVector, V::CuVector, [m, n]; fmt=:csc)
+
+Return a sparse cuda matrix, with type determined by `fmt`.
+Possible formats are :csc, :csr, :bsr, and :coo.
+"""
+function SparseArrays.sparse(x::DenseCuMatrix; fmt=:csc)
+    if fmt == :csc
+        return CuSparseMatrixCSC(x)
+    elseif fmt == :csr
+        return CuSparseMatrixCSR(x)
+    elseif fmt == :bsr
+        return CuSparseMatrixBSR(x)
+    elseif fmt == :coo
+        return CuSparseMatrixCOO(x)
+    else
+        error("Format :$fmt not available, use :csc, :csr, :bsr or :coo.")
+    end
+end
+
+SparseArrays.sparse(I::CuVector, J::CuVector, V::CuVector; kws...) =
+    sparse(I, J, V, maximum(I), maximum(J); kws...)
+
+SparseArrays.sparse(I::CuVector, J::CuVector, V::CuVector, m, n; kws...) =
+    sparse(Cint.(I), Cint.(J), V, m, n; kws...)
+
+function SparseArrays.sparse(I::CuVector{Cint}, J::CuVector{Cint}, V::CuVector{Tv}, m, n;
+            fmt=:csc) where Tv
+    spcoo = CuSparseMatrixCOO{Tv}(I, J, V, (m, n))
+    if fmt == :csc
+        return CuSparseMatrixCSC(spcoo)
+    elseif fmt == :csr
+        return CuSparseMatrixCSR(spcoo)
+    elseif fmt == :coo
+        return spcoo
+    else
+        error("Format :$fmt not available, use :csc, :csr, or :coo.")
+    end
+end
 
 
 ## CSR to CSC
@@ -238,6 +277,14 @@ function CuSparseMatrixCOO(csr::CuSparseMatrixCSR{Tv}, ind::SparseChar='O') wher
     CuSparseMatrixCOO{Tv}(cooRowInd, csr.colVal, csr.nzVal, csr.dims, nnz)
 end
 
+### CSC/BST to COO and viceversa
+
+CuSparseMatrixCSC(coo::CuSparseMatrixCOO) = CuSparseMatrixCSC(CuSparseMatrixCSR(coo)) # no direct conversion
+CuSparseMatrixCOO(csc::CuSparseMatrixCSC) = CuSparseMatrixCOO(CuSparseMatrixCSR(csc)) # no direct conversion
+CuSparseMatrixBSR(coo::CuSparseMatrixCOO, blockdim) = CuSparseMatrixBSR(CuSparseMatrixCSR(coo), blockdim) # no direct conversion
+CuSparseMatrixCOO(bsr::CuSparseMatrixBSR) = CuSparseMatrixCOO(CuSparseMatrixCSR(bsr)) # no direct conversion
+
+
 ## sparse to dense, and vice-versa
 
 for (cname,rname,elty) in ((:cusparseScsc2dense, :cusparseScsr2dense, :Float32),
diff --git a/test/cusparse/conversions.jl b/test/cusparse/conversions.jl
new file mode 100644
index 0000000000..83190338d8
--- /dev/null
+++ b/test/cusparse/conversions.jl
@@ -0,0 +1,36 @@
+using CUDA.CUSPARSE, SparseArrays
+
+@testset "sparse" begin
+    n, m = 4, 4
+    I = [1,2,3] |> cu
+    J = [2,3,4] |> cu
+    V = Float32[1,2,3] |> cu
+
+    dense = rand(3,3) |> cu
+
+    # check defaults
+    @test sparse(I, J, V) isa CuSparseMatrixCSC
+    @test sparse(dense) isa CuSparseMatrixCSC
+
+    for (fmt, T) in  [(:coo, CuSparseMatrixCOO),
+                      (:csc, CuSparseMatrixCSC),
+                      (:csr, CuSparseMatrixCSR),
+                      (:bsr, CuSparseMatrixBSR)
+                     ]
+        if fmt != :bsr # bsr not supported
+            x = sparse(I, J, V; fmt=fmt)
+            @test x isa T{Float32}
+            @test size(x) == (3, 4)
+
+            x = sparse(I, J, V, m, n; fmt=fmt)
+            @test x isa T{Float32}
+            @test  size(x) == (4, 4)
+        end
+
+        if fmt != :coo # dense to COO not implemented
+            x = sparse(dense; fmt=fmt)
+            @test x isa T{Float32}
+            @test collect(x) == collect(dense)
+        end
+    end
+end