Last decade has witnessed a rapid development of the generation of terahertz (THz) vortex beams as well as their wide applications, mainly due to their unique combination characteristics of regular THz radiation and orbital angular momentum (OAM). Here we have reviewed the ways to generate THz vortex beams by two representative scenarios, i.e., THz wavefront modulation via specific devices, and direct excitation of the helicity of THz vortex beams. The former is similar to those wavefront engineering devices in the optical and infrared (IR) domain, but just with suitable THz materials, while the latter is newly-developed in THz regime and some of the physical mechanisms still have not been explained explicitly enough though, which would provide both challenges and opportunities for THz vortex beam generation. As for their applications, thanks to the recent development of THz optics and singular optics, THz vortex beams have potentials to open doors towards a myriad of practice applications in many fields. Besides, some representative potential applications are evaluated such as THz wireless communication, THz super-resolution imaging, manipulating chiral matters, accelerating electron bunches, and detecting astrophysical sources.