With the rapid development of robotics, it is more and more difficult for traditional actuators with redundant structures and large sizes (e.g., motors, hydraulic actuators, etc.) to meet the needs of lightweight and flexibility for the new generation of intelligent robots. Hence, pneumatic artificial muscle (PAM) actuators with higher flexibility and safety have been attracting more and more attention in recent years. The PAM is simple in structure, light in material, and good in biological adaptability. It has good adaptability in medical rehabilitation, aerospace, underwater operations, emergency relief and other fields, which can be easily used to actuate robots to complete a number of complex tasks. However, PAMs have inherent hysteresis, high nonlinearities, and creep characteristics, which bring challenges for accurate dynamic modeling and controller design for flexible robots actuated by PAMs. In this paper, first, we briefly introduce the PAM's working principle, advantages and disadvantages, modeling methods, and related applications. Then, based on the mainstream model of PAMs, the research status and latest progress of motion control methods for single and multi-PAM-driven robots in recent years are emphatically described. Finally, according to the current research progress and unsolved issues, future development trends of flexible robots driven by PAMs are summarized.