  Topological insulators are characterized by insulating bulk and conducting
surface, the latter is a necessity consequence of the nontrivial topology of
the wavefunctions forming the valence band. This chapter gives a historical
overview of the discovery of topological insulators and a concise description
of the $Z_2$ topology which defines them. The concept of topological insulators
have been extended to various other topologies, giving rise to the recognition
of further topological states of matter such as topological crystalline
insulators and higher-order topological insulators. Representative materials of
topological insulators, their synthesis techniques, and the ways for the
experimental confirmation of the topological nature are introduced. Among the
interesting phenomena derived from topological insulators, topological
superconductivity, Majorana zero modes, and quantum anomalous Hall effect are
briefly discussed.
