Floquet systems are governed by periodic, time-dependent Hamiltonians. Prima facie they should absorb energy from the external drives involved in modulating their couplings and heat up to infinite temperature. However, this unhappy state of affairs can be avoided in many ways. Instead, as has become clear from much recent work, Floquet systems can exhibit a variety of nontrivial behavior—some of which is impossible in undriven systems. In this review, we describe the main ideas and themes of this work: novel Floquet drives that exhibit nontrivial topology in single-particle systems, the existence and classification of exotic Floquet drives in interacting systems, and the attendant notion of many-body Floquet phases and arguments for their stability to heating.