The topology of electronic band structures

The study of topology as it relates to physical systems has rapidly accelerated during the past decade. Critical to the realization of new topological phases is an understanding of the materials that exhibit them and precise control of the materials chemistry. The convergence of new theoretical methods using symmetry indicators to identify topological material candidates and the synthesis of high-quality single crystals plays a key role, warranting discussion and context at an accessible level. This Perspective provides a broad introduction to topological phases, their known properties, and material realizations. We focus on recent work in topological Weyl and Dirac semimetals, with a particular emphasis on magnetic Weyl semimetals and emergent fermions in chiral crystals and their extreme responses to excitations, and we highlight areas where the field can continue to make remarkable discoveries. We further examine open questions and directions for the topological materials science community to pursue, including exploration of non-equilibrium properties of Weyl semimetals and cavity-dressed topological materials.