I will discuss how non-singular bouncing cosmologies can lead to an alternative to cosmological inflation for the origin of structure in the universe. However, in order to obtain a non-singular bounce, physics which goes beyond General Relativity coupled to matter satisfying the usual energy conditions is required. I will discuss two ways of achieving a non-singular bounce, one obtained by modifying the matter sector of the theory (ghost condensate bounce), the other by modifying the gravitational action (Horava-Lifshitz bounce).