I will discuss how topological phases arise in quantum matter through spin-orbit coupling effects in the presence ofprotectionsprovided by time-reversal, crystalline and particle-hole symmetries, and highlight our recent work aimed at predicting newclasses of topological insulators (TIs), topological crystalline insulators, Weyl semi-metals, and quantum spin Hall insulators.[1-10] Surfaces of three-dimensional (3D) topological materials and edges of two-dimensional (2D) topological materials supportnovel electronic states. For example, the surface of a 3D TI supports gapless or metallic states, which are robust against disorderand non-magnetic impurities, and in which the directions of momentum and spin are locked with each other. Similarly, in2D TIs, also called quantum spin Hall insulators, the 1D topological edge states are not allowed to scatter since the only availablebackscattering channel is forbidden by constraints of time-reversal symmetry. The special symmetry protected electronic statesin topological materials hold the exciting promise of providing revolutionary new platforms for exploring fundamental sciencequestions, including novel spin textures and exotic superconductors, and for the realization of multifunctional topological devicesfor thermoelectric, spintronics, information processing and other applications. Work supported by the U. S. Departmentof Energy.