Space robots play a significant role in on‐orbit servicing (OOS) missions, such as inspecting, capturing, refueling, and repairing satellites, assembling and maintaining large space infrastructure, and removing orbital debris. Over the past four decades, many space robot engineering applications and technology verifications for OOS have been accomplished. This article comprehensively reviews the advances by representative space robotic programs on space shuttles, outside/inside the International Space Station and the China Space Station, as well as on satellites, and the development trends of space robots are summarized. In addition, the primary key technologies and challenges are explored, including the following: 1) visual perception for noncooperative targets; 2) motion planning and control with a free‐floating base and flexibility; 3) multifunctional end‐effectors; 4) ground teleoperation with long time delays; and 5) high‐fidelity ground verification. Finally, the prospects for space robot future research are presented.