With the accelerated pace of human space exploration and the progress of other related researches, there is an increasingly urgent demand for space infrastructure, equipment, and diversified spacecraft construction for space missions, and how to efficiently, intelligently, and autonomously build corresponding facilities and equipment on orbit according to the functional requirements of different missions has become a great challenge in the field of space technology research. As an important means of automated manufacturing, the construction of on-orbit assembly systems centered on space robotics has become an emerging development trend. In view of its importance, space agencies and research institutes have successively proposed and developed a series of related programs. In order to comprehensively understand the progress of on-orbit assembly with space robots (OASR) and scientific problems involved, this paper investigates the current status of research and technological development in OASR. Firstly, the significance of OASR for space exploration and other space missions is analyzed. Secondly, the existing classification forms of on-orbit assembly are outlined and a classification idea is proposed from the point of view of the combination of space robot motion capability and assembly goals. Thirdly, the research and development status of OASR in the United States, Europe, Canada, Japan, and China is investigated. Then, based on a review of the literature on space robots to realize on-orbit assembly in space facilities, some of the key technologies involved are reviewed and discussed. Finally, this paper discusses and looks ahead to the future development trend and application prospect of the technology of OASR, reveals and explains the crucial position it occupies as well as the important role it can play in the process of human space exploration, and is expected to provide useful references for the in-depth research and development of future on-orbit assembly technology.