The piezoelectric-actuated compliant microgripper (PEACM) plays an essential role in the application fields such as biomedical engineering, microelectronics, and optical engineering. As compared with other categories of grippers, PEACM exhibits the advantages of high accuracy of displacement, large power to weight ratio, low energy consumption, and fast response speed. This paper reviews the recent advances on performance indices, classification, structural composition, optimization and modeling method, and control of PEACM. First, the gripper's performance indices and classifications are elaborated, which is beneficial to determine the design goal. Then, the compliant mechanisms adopted in the microgripper design are discussed, including the flexible hinge, displacement amplifier, and guiding mechanism. In addition, the optimization and modeling methods of the microgripper are presented. Popular types of position/force sensors and different displacement/force control strategies employed in the microgripper are surveyed. Moreover, the prospect on future development trend of the PEACM is discussed. The paper provides the reader with an overview of the recent advances on PEACM design and also a guideline on further development of the microgripper.