  We address the existence and stability of two-dimensional solitons in optical
or matter-wave media, which are supported by purely nonlinear lattices in the
form of a periodic array of cylinders with self-focusing nonlinearity, embedded
into a linear material. We show that such lattices can stabilize
two-dimensional solitons against collapse. We also found that stable multipoles
and vortex solitons are also supported by the nonlinear lattices, provided that
the nonlinearity exhibits saturation.
