  Topological materials discovery has evolved at a rapid pace over the past 15
years following the identification of the first nonmagnetic topological
insulators (TIs), topological crystalline insulators (TCIs), and 3D topological
semimetals (TSMs). Most recently, through complete analyses of symmetry-allowed
band structures - including the theory of Topological Quantum Chemistry (TQC) -
researchers have determined crystal-symmetry-enhanced Wilson-loop and complete
symmetry-based indicators for nonmagnetic topological phases, leading to the
discovery of higher-order TCIs and TSMs. The recent application of TQC and
related methods to high-throughput materials discovery has revealed that over
half of all of the known stoichiometric, solid-state, nonmagnetic materials are
topological at the Fermi level, over 85% of the known stoichiometric materials
host energetically isolated topological bands, and that just under $2/3$ of the
energetically isolated bands in known materials carry the stable topology of a
TI or TCI. In this Review, we survey topological electronic materials discovery
in nonmagnetic crystalline solids from the prediction of the first 2D and 3D
TIs to the recently introduced methods that have facilitated large-scale
searches for topological materials. We also discuss future venues for the
identification and manipulation of solid-state topological phases, including
charge-density-wave compounds, magnetic materials, and 2D few-layer devices.
