  Clean and interacting periodically driven quantum systems are believed to
exhibit a single, trivial "infinite-temperature" Floquet-ergodic phase. In
contrast, here we show that their disordered Floquet many-body localized
counterparts can exhibit distinct ordered phases delineated by sharp
transitions. Some of these are analogs of equilibrium states with broken
symmetries and topological order, while others - genuinely new to the Floquet
problem - are characterized by order and non-trivial periodic dynamics. We
illustrate these ideas in driven spin chains with Ising symmetry.
