A bioinspired jumping robot has a strong ability to overcome obstacles. It can be applied to the occasion with complex and changeable environment, such as detection of planet surface, postdisaster relief, and military reconnaissance. So the bioinspired jumping robot has broad application prospect. The jumping process of the robot can be divided into three stages: takeoff, air posture adjustment, and landing buffer. The motivation of this review is to investigate the research results of the most published bioinspired jumping robots for these three stages. Then, the movement performance of the bioinspired jumping robots is analyzed and compared quantitatively. Then, the limitation of the research on bioinspired jumping robots is discussed, such as the research on the mechanism of biological motion is not thorough enough, the research method about structural design, material applications, and control are still traditional, and energy utilization is low, which make the robots far from practical applications. Finally, the development trend is summarized. This review provides a reference for further research of bioinspired jumping robots.