  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, they can exhibit a variety of nontrivial behavior---some of it
impossible in undriven systems. In this review we describe the main ideas and
themes of this work: novel Floquet drives which 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.
