Ferromagnet/nonmagnet (FM/NM) heterostructure under the excitation of femtosecond laser has proved to be a potential candidate for high-efficiency terahertz (THz) emission. Topological insulator (TI) is a novel two-dimensional (2D) material with a strong spin-orbital coupling, which endows this material with an extremely large spin-Hall angle. Thus, TI appears to be an attractive alternative to achieving higher-performance spintronic THz emitter when integrated with ferromagnetic material. In this paper, we discuss the ultrafast photocurrent response mechanism in TI film on the basis of the analysis of its crystal and band structures. The discussion of the mechanism reveals a relationship between THz radiation and external conditions, such as crystal orientation, polarized direction and chirality of the laser. Furthermore, we review the spintronic THz emission and manipulation in FM/NM heterostructure. The disclosed relationship between THz radiation and magnetization directions enables an effective control of the THz polarization by optimizing the system, such as by applying twisted magnetic field or fabricating cascade emitters. After integration, the FM/TI heterostructure presents a high efficiency and easy operation in THz radiation. This high-performance topological spintronic THz emitter presents a potential for the achievement of arbitrary polarization-shaping terahertz radiation.