  We consider effects of anisotropy on solitons of various types in
two-dimensional nonlinear lattices, using the discrete nonlinear
Schr{\"{o}}dinger equation as a paradigm model. For fundamental solitons, we
develop a variational approximation, which predicts that broad quasi-continuum
solitons are unstable, while their strongly anisotropic counterparts are
stable. By means of numerical methods, it is found that, in the general case,
the fundamental solitons and simplest on-site-centered vortex solitons ("vortex
crosses") feature enhanced or reduced stability areas, depending on the
strength of the anisotropy. More surprising is the effect of anisotropy on the
so-called "super-symmetric" intersite-centered vortices ("vortex squares"),
with the topological charge $S$ equal to the square's size $M$: we predict in
an analytical form by means of the Lyapunov-Schmidt theory, and confirm by
numerical results, that arbitrarily weak anisotropy results in dramatic changes
in the stability and dynamics in comparison with the \emph{degenerate}, in this
case, isotropic limit.
