Abstract: We demonstrate existence of discrete solitons in Discrete Nonlinear Schrodinger equation (dlns) with saturable nonlinearity. We consider two types of solutions to (dlns) periodic and vanishing at infinity. In the second part of our talk, we prove the existence of periodic and solitary traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. Calculus of variations and Nehari manifolds are employed to establish the existence of these solutions. We present some extensions of our results, combining the Nehari manifold approach and the Mountain Pass argument.