  Different ways to incorporate two-dimensional systems, which are not amenable
to separation of variables, into the framework of Supersymmetrical Quantum
Mechanics (SUSY QM) are analyzed. In particular, the direct generalization of
one-dimensional Witten's SUSY QM is based on the supercharges of first order in
momenta and allows to connect the eigenvalues and eigenfunctions of two scalar
and one matrix Schr\"odinger operators. The use of second order supercharges
leads to polynomial supersymmetry and relates a pair of scalar Hamiltonians,
giving a set of such partner systems with almost coinciding spectra. This class
of systems can be studied by means of new method of $SUSY-$separation of
variables, where supercharges {\bf allow} separation of variables, but
Hamiltonians {\bf do not}. The method of shape invariance is generalized to
two-dimensional models to construct purely algebraically a chain of eigenstates
and eigenvalues for generalized Morse potential models in two dimensions.
