Space exploration and exploitation depend on the development of on-orbit robotic capabilities for tasks such as servicing of satellites, removing of orbital debris, or construction and maintenance of orbital assets. Manipulation and capture of objects on-orbit are key enablers for these capabilities. This survey addresses fundamental aspects of manipulation and capture, such as the dynamics of space manipulator systems (SMS), i.e., satellites equipped with manipulators, the contact dynamics between manipulator grippers/payloads and targets, and the methods for identifying properties of SMSs and their targets. Also, it presents recent work of sensing pose and system states, of motion planning for capturing a target, and of feedback control methods for SMS during motion or interaction tasks. Finally, the paper reviews major ground testing testbeds for capture operations, and several notable missions and technologies developed for capture of targets on-orbit.