  We apply the combinatorial theory of spherical varieties to characterize the
momentum polytopes of polarized projective spherical varieties. This enables us
to derive a classification of these varieties, without specifying the open
orbit, as well as a classification of all Fano spherical varieties. In the
setting of multiplicity free compact and connected Hamiltonian manifolds, we
obtain a necessary and sufficient condition involving momentum polytopes for
such manifolds to be K\"ahler and classify the invariant compatible complex
structures of a given K\"ahler multiplicity free compact and connected
Hamiltonian manifold.
