Abstract: Exploring the topologically nontrivial states in condensed matter, especially their electronic structure and transport property, is very important and timely. Most topics discussed in this paper are relevant to the topological properties of electronic wavefunctions. Besides a brief introduction, the paper contains five sections, including topological quantum phenomena, various topological phases, unusual transport properties of topological quasiparticles, topological collective excitations and coupled excitations, and ongoing research in topological quantum states. These presentations expound a variety of aspects in the studies of topological quantum states. With all these aspects together, a comprehensive understanding of the topological quantum states in condensed matter physics can be achieved.

Key words: Quantum Hall effect; Berry phase; Topological insulator; Topological semimetal; Topological superconductor; Dirac electron; Topological excitation; Quantum transport