Complex precision components are integral to many sectors, straddling both military and civilian applications. These include aerospace telescopes, infrared thermal imaging systems, artificial intelligence, semiconductor chip lithography, medical imaging apparatus, and avant-garde communication technologies. These intricate precision components have become vital elements of the aforementioned optical systems, characterized by a wide range of extensive requirements totaling in the tens of millions. Within the realm of computer controlled optical surfacing (CCOS), high-efficiency bonnet polishing (BP) and high-precision magnetorheological finishing (MRF) are two compliant polishing methods with distinct advantages, extensively applied to ultra-precision machining of complex curved surface components. However, the bonnet polishing tool is prone to wear, the tool influence function is unstable, and the control process is complicated. The material removal efficiency of MRF is low; it easily introduces mid-spatial frequency (MSF) errors, and improving the performance of the magnetorheological fluid (MR fluid) is challenging. Therefore, summarizing these two techniques is essential to enhance the application of compliant polishing methods. The paper begins by examining the unique strengths of both technologies and then explores the potential for their integrated application. The paper then provides a detailed introduction to the origin, principles, equipment, and applications of BP. Next, the paper outlines the research progress of key technologies, including modeling of the tool influence function (TIF), management of MSF errors, and the wear of the bonnet tool within the realm of BP technology. Following that, the development history, technical principles, equipment, types, and compound methods of MRF are presented. Then, the research progress of several key technologies, such as modeling of TIF, controlling MSF error, and the preparation of MR fluid in the field of MRF technology, are reviewed. Lastly, the paper provides a summary and outlook for the two technologies, such as further in-depth study of the material removal mechanism and the suppression method of the edge effect in BP, a further in-depth study of methods to improve the material removal rate, and MSF error suppression methods in MRF.