  The first-principles band theory paradigm has been a key player not only in
the process of discovering new classes of topologically interesting materials,
but also for identifying salient characteristics of topological states,
enabling direct and sharpened confrontation between theory and experiment. We
begin this review by discussing underpinnings of the topological band theory,
which basically involves a layer of analysis and interpretation for assessing
topological properties of band structures beyond the standard band theory
construct. Methods for evaluating topological invariants are delineated,
including crystals without inversion symmetry and interacting systems. The
extent to which theoretically predicted properties and protections of
topological states have been verified experimentally is discussed, including
work on topological crystalline insulators, disorder/interaction driven
topological insulators (TIs), topological superconductors, Weyl semimetal
phases, and topological phase transitions. Successful strategies for new
materials discovery process are outlined. A comprehensive survey of currently
predicted 2D and 3D topological materials is provided. This includes binary,
ternary and quaternary compounds, transition metal and f-electron materials,
Weyl and 3D Dirac semimetals, complex oxides, organometallics, skutterudites
and antiperovskites. Also included is the emerging area of 2D atomically thin
films beyond graphene of various elements and their alloys, functional thin
films, multilayer systems, and ultra-thin films of 3D TIs, all of which hold
exciting promise of wide-ranging applications. We conclude by giving a
perspective on research directions where further work will broadly benefit the
topological materials field.
