We describe recent progress in our understanding of the interplay between
interactions, symmetry, and topology in states of quantum matter. We focus on a
minimal generalization of the celebrated topological band insulators to
interacting many particle systems, known as Symmetry Protected Topological
(SPT) phases. In common with the topological band insulators these states have
a bulk gap and no exotic excitations but have non-trivial surface states that
are protected by symmetry. We describe the various possible such phases and
their properties in three dimensional systems with realistic symmetries. We
develop many key ideas of the theory of these states using simple examples. The
emphasis is on physical rather than mathematical properties. We survey insights
obtained from the study of SPT phases for a number of other theoretical
problems.Comment: Invited Review for Annual Reviews of Condensed Matter Physic
