Periodic driving can be used to induce a wide array of interesting phenomena in quantum many-body systems.  In addition to providing means to induce artificial gauge fields or to realize familiar effective Hamiltonians, periodic driving opens a wide new world of intrinsically non-equilibrium quantum phases. Intriguingly, the topological classification of periodically-driven systems is more rich than that in equilibrium, allowing for a variety of "anomalous Floquet insulator" phases which feature characteristic patterns of nontrivial micromotion within each driving period.

In this talk I will give an introduction to the unique features of topology in periodically-driven systems, and discuss the new quantized observables that can be used to detect them. In particular, I will focus on a two-dimensional system that features chiral edge states even though all bulk bands have vanishing Chern numbers and may be fully localized by disorder.  This phase features a quantized orbital magnetization density in any filled region, which serves as a bulk topological order parameter for the phase (and applies in both interacting and non-interacting systems). Due to its chiral edge states, which are not localized by disorder, this phase provides an interesting platform for studying the interplay between thermalization and many-body localization.