In recent years, by introducing topological invariants into condensed matter systems, new phases of mater are revealed. Of these new phases, the topological insulator, topological semimetal and topological superconductor are the most important. They are called topological materials due to nontrivial topological parameters. Topological superconductors hold Majorana zero modes at the edges, satisfying non-abelian statistics, which makes them major candidate for realizing topological quantum computation. Besides exploring intrinsic topological superconductor, a promising way to realize topological superconductor is to induce superconductivity into other kinds of topological materials. Up to now, experimentalists have developed some techniques, such as gating, doping, high pressure, interface effect and hard point contact to introduce superconductivity into various topological materials, and also they have studied the topological properties of the induced superconductivity. In this review, we summarize the representative researches on intrinsic topological superconductor candidates and induced superconductivities in topological insulators and semimetals. The advantages and disadvantages of different techniques are discussed. Besides, the potential evidences of topological superconductors are analyzed. In the end, the outlook of this actively pursued research field is given.