  The discovery of the quantum spin Hall effect and topological insulators more
than a decade ago has revolutionized modern condensed matter physics. Today,
the field of topological states of matter is one of the most active and
fruitful research areas for both experimentalists and theorists. The physics of
topological insulators is typically well described by band theory and systems
of non-interacting fermions. In contrast, several of the most fascinating
effects in condensed matter physics merely exist due to electron-electron
interactions, examples include unconventional superconductivity, the Kondo
effect, and the Mott-Hubbard transition. The aim of this review article is to
give an overview of the manifold directions which emerge when topological
bandstructures and correlation physics interfere and compete. These include the
study of the stability of topological bandstructures and correlated topological
insulators. Interaction-induced topological phases such as the topological
Kondo insulator provide another exciting topic. More exotic states of matter
such as topological Mott insulator and fractional Chern insulators only exist
due to the interplay of topology and strong interactions and do not have any
bandstructure analogue. Eventually the relation between topological
bandstructures and frustrated quantum magnetism in certain transition metal
oxides is emphasized.
