  Topological materials have become the focus of intense research in recent
years, since they exhibit fundamentally new physical phenomena with potential
applications for novel devices and quantum information technology. One of the
hallmarks of topological materials is the existence of protected gapless
surface states, which arise due to a nontrivial topology of the bulk wave
functions. This review provides a pedagogical introduction into the field of
topological quantum matter with an emphasis on classification schemes. We
consider both fully gapped and gapless topological materials and their
classification in terms of nonspatial symmetries, such as time-reversal, as
well as spatial symmetries, such as reflection. Furthermore, we survey the
classification of gapless modes localized on topological defects. The
classification of these systems is discussed by use of homotopy groups,
Clifford algebras, K-theory, and non-linear sigma models describing the
Anderson (de-)localization at the surface or inside a defect of the material.
Theoretical model systems and their topological invariants are reviewed
together with recent experimental results in order to provide a unified and
comprehensive perspective of the field. While the bulk of this article is
concerned with the topological properties of noninteracting or mean-field
Hamiltonians, we also provide a brief overview of recent results and open
questions concerning the topological classifications of interacting systems.
