Periodically driven (Floquet) quantum systems can exhibit new non-equilibrium phases of matter that are intrinsically dynamical and could not occur in a static systems. In this talk, I will examine new forms of symmetry protected topological (SPT), symmetry enriched topological (SET) phases, and quantum criticality that arise in Floquet systems, with an emphasis on systems with strong interactions. I will describe the prospects for exploring these new phenomena in cold-atoms and trapped ion systems.