Most of the atoms that realize Bose-Einstein condensation have internal spin degree of freedom. In the optical potential trap, the internal spin of the atom is thawed, and the atom can be condensed into each hyperfine quantum state to form the spinor Bose-Einstein condensate. Flexible spin degrees of freedom become dynamic variables related to the system, which can make the system appear novel topological quantum states, such as spin domain wall, vortex, magnetic monopole, skymion, and so on. In this paper, the experimental and theoretical study of spinor Bose-Einstein condensation, the types of topological defects in spinor Bose-Einstein condensate, and the research progress of topological defects in spinor two-component and three-component Bose-Einstein condensate are reviewed.