Cable-driven robots are driven by flexible cables instead of traditional rigid links, significantly reducing the weight of the mechanism and the inertia of the moving components, breaking through the limitations of the hinge angle and telescopic length, and offering obvious advantages in terms of workspace, load-to-weight ratio, and adaptability. The development of cable-driven robots represents a lightweight design concept and a development trend of human-robot cooperation. In this review, the representative theories, typical applications and research advancements of cable-driven robots are discussed in areas such as motion planning, trajectory tracking control, tension distribution strategies, vibration suppression and fault-tolerant control. The design approaches and advantages of various motion planning and control methods are analyzed and the challenges in each area are concluded. Finally, the development trends for motion planning and control methods of cable-driven robots are summarized in three aspects, including research on autonomous reconfiguration, variable stiffness control and natural interaction and human-robot cooperative control.