In order to clarify the cavity design methods of thin-disk multi-pass amplifiers, we summarize the different types of thin-disk multi-pass amplifiers and concludes that there are four fundamental design concepts: (1) 4f relay imaging, (2) resonant cavity design/optical Fourier transform, (3) near-collimated beam transmission, and (4) others. Each amplifier design method is described and the current status of its research is listed in as much detail as possible. By comparing the four types of disk multi-pass amplifiers, it is found that the varying methods have distinct advantages and disadvantages. 4f relay imaging requires a vacuum environment to avoid gas ionization at the focal point, making the mechanics and adjustment more difficult; the resonant cavity design/optical Fourier transform concept multi-pass amplifier has a small spot at the mirrors, making it more suitable for lower energy multi-pass amplifiers; the near collimated beam transmission method has great development potential because it does not require a vacuum environment, but accurately controlling the surface shape of the thin-disk is difficult while the laser is operating. Therefore, from the perspective of laser design, it is necessary to continue to optimize the design of the thin-disk multi-pass amplifier to realize the diversification of application scenarios and the sustainable expansion of output energy.