As the low-altitude economy enters a phase of rapid development, rotorcraft unmanned aerial vehicles (UAVs) have attracted widespread attention due to their vertical takeoff and landing capabilities, high maneuverability and low cost. The utilization of it for aerial transportation of materials and equipment offers a significant advantage by overcoming the constraints imposed by challenging terrains, thereby providing a more efficient alternative to traditional ground transportation methods. In the field of aerial transportation, the cable-suspend method offers advantages such as low cost and no restrictions on payload volume, making it highly adaptable for various missions. However, rotorcraft UAVs are inherently underactuated, and integrating the payload further increases the system's degrees of freedom, exacerbating the nonlinearity and state coupling. These complexities present significant challenges to achieving effective system control and motion planning. This paper first classifies the cable-suspended payload transportation systems by rotorcraft UAVs and briefly introduces their characteristics. Then, it focuses on the current state of research and recent advancements in the control and planning methods for both single-UAV and multi-UAV systems. Finally, the existing research in the field of cable-suspended payload transportation systems by rotorcraft UAVs is discussed and summarized, and the future developments and trends in this area are anticipated.