Yttrium iron garnet (Y3Fe5O12, YIG) has emerged as a highly promising functional material for next-generation optical isolators, magneto-optical switches, and integrated photonic devices, owing to its excellent optical transparency, strong Faraday rotation, low optical absorption loss, and high Curie temperature in the near- to mid-infrared spectral region. The rapid advancement of 5G/6G optical communications, fiber-optic gyroscopes, and quantum computing imposes increasingly stringent demands for high-performance, low-power magneto-optical devices, which in turn require YIG single crystals to combine a large magneto-optical response with low insertion loss and robust temperature stability. However, the incongruent melting behavior of YIG precludes the growth of high-quality single crystals by conventional Czochralski pulling, making the flux method the critical technological route for YIG crystal fabrication. This review provides a systematic overview of the evolution of flux-based growth techniques for YIG single crystals, with emphasis on the selective occupation mechanisms at the dodecahedral c-site and the associated charge-compensation principles of Bi3+ and representative rare-earth ions such as Tb3+, Ce3+, Gd3+, and Dy3+. From the perspective of the microscopic origins of the magneto-optical effect, the incorporation of these ions enhances the Faraday rotation predominantly by strengthening spin-orbit coupling, introducing new 4f-5d transition channels, or modifying the super-exchange interactions among the Fe3+ sublattices. Building on these fundamentals, this review further discusses the strategies for tailoring the magneto-optical performance in the near- to mid-infrared region and for optimizing temperature stability through ion doping. Current technical challenges, including doping homogeneity, crystal defect control, and broadband magneto-optical response, are analyzed, and future directions toward integrated-photonics-oriented magneto-optical single crystals are outlined.