The monitoring requirements of temperature, strain, and other physical parameters during the assembling, cooling, and operation of low-temperature and high-temperature superconducting magnets are introduced. The working principles of fiber-optic sensing technologies based on the fiber Bragg grating, Brillouin scattering, Raman scattering, and Rayleigh scattering are also introduced. The research progress of the above optical fiber sensing technologies in temperature and strain measurement of low-temperature and high-temperature superconducting magnets is reviewed. The existing problems in the above researches are summarized, and the future research is prospected. This review is helpful for researchers in the field of superconducting magnet, especially in the field of superconducting magnet stability, to understand the application progress of optical fiber sensing technology in superconducting magnet stability improvement intuitively, clearly, and comprehensively.