  The problem of classification of decomposable (in the sense of Stormer)
positive maps between matrix algebras is presented. We propose the new notion
of "finite" version of decomposability ($k$-decomposabilty). The
characterisation of $k$-decomposability on the Hilbert space level is done. In
the case of low dimensional algebras the notion of local decomposability and
its applications for the description of decomposable maps are discussed.
