Abstract：Topological materials are solids with nontrivial topology in their electronic band structures. The periodical lattice of atoms in solids limits the electrons in a compact reciprocal lattice space and leads to nontrivial band topology in both insulators and metals, namely topological insulators and topological semimetals. Band inversion is an intuitive picture for understanding the underlying physics since it is related with Berry phase and Berry curvature of bands. How to find a material with nontrivial band topology had been a quite elusive and difficult task. However, some exotic and abnormal physical phenomena related with nontrivial band topology have been noticed and successfully applied for locating the exact topological materials, which includes a large band gap two-dimensional TIs in ZrTe5, the first Dirac semimetal Na3Bi and the first Weyl semimetal TaAs. These discoveries have greatly advanced the whole field of topological quantum states. Recently, the band inversion have been found to be efficiently identified with symmetry indicators formed by irreducible representation of bands at high symmetrical momenta, which can be obtained from first-principles band structure calculations. High-throughput calculation of known non-magnetic materials is thus performed and thousands compounds have been identified to have nontrivial band topology, which is put into a database for routinely search.