Iron-based superconductors and topological quantum states have been two important research frontiers in condensed matter physics in recent years. It is a very significant question whether the nontrivial topological phenomena can occur in iron-based superconductors. In this paper, the basic characteristics of the electronic structure of iron-based superconducting are analyzed from three aspects:crystal symmetry, effective model near the high symmetry points in Brillouin zone, and spin-orbit coupling interaction. On this basis, this paper focuses on how the nontrivial topological quantum states occur in the normal state, the states with long-range order near superconducting state and the superconducting state. Furthermore, the relevant theoretical models and results are introduced in detail, the related experimental progress is reviewed, and the development in this field is prospected.