  Topology is now securely established as a means to explore and classify
electronic states in crystalline solids. This review provides a gentle but firm
introduction to topological electronic band structure suitable for new
researchers in the field. I begin by outlining the relevant concepts from
topology, then give a summary of the theory of non-interacting electrons in
periodic potentials. Next, I explain the concepts of the Berry phase and Berry
curvature, and derive key formulae. The remainder of the article deals with how
these ideas are applied to classify crystalline solids according to the
topology of the electronic states, and the implications for observable
properties. Among the topics covered are the role of symmetry in determining
band degeneracies in momentum space, the Chern number and Z2 topological
invariants, surface electronic states, two- and three-dimensional topological
insulators, and Weyl and Dirac semimetals
