  We study the existence, stability, and mobility of fundamental discrete
solitons in two- and three-dimensional nonlinear Schroedinger lattices with a
combination of cubic self-focusing and quintic self-defocusing onsite
nonlinearities. Several species of stationary solutions are constructed, and
bifurcations linking their families are investigated using parameter
continuation starting from the anti-continuum limit, and also with the help of
a variational approximation. In particular, a species of hybrid solitons,
intermediate between the site- and bond-centered types of the localized states
(with no counterpart in the 1D model), is analyzed in 2D and 3D lattices. We
also discuss the mobility of multi-dimensional discrete solitons that can be
set in motion by lending them kinetic energy exceeding the appropriately
crafted Peierls-Nabarro barrier; however, they eventually come to a halt, due
to radiation loss.
