Recently, distributed Brillouin optical fiber sensors have been extensively studied and discussed for the capacity to measure distributed strain and temperature, as well as important applications in the field of structural health monitoring. In several optical fiber sensing schemes, Brillouin optical time domain analysis (BOTDA) is widely concerned due to its good signal-to-noise ratio, high spatial resolution, and long-range sensing distance. However, due to the time-consuming averaging and frequency-sweeping processes, the classical BOTDA systems are suitable for static or slow-varying strain measurements. In this paper, we analyze the operation principle of BOTDA system and discuss some main factors for limiting fast measurement. Then, we summarize and analyze the dynamic measurement methods based on fast BOTDA, which are polarization compensation technique, frequency-agile technique, slope-assisted method, optical chirp chain technique, optical frequency comb technique. It is pointed out that the fast BOTDA system consisted of a single or multiple new techniques, has a better performance and a wider application prospect.