This study focuses on the flapping mechanisms found in recently developed biometric flapping-wing air vehicles (FWAVs). FWAVs mimic the flight characteristics of flying animals, providing advantages such as maneuverability, inconspicuousness, and excellent flight efficiency in the low Reynolds number region. The flapping mechanism is a critical part of determining the aerodynamic performance of an FWAV since it is directly related to the wing motion. In this study, the flight characteristics of birds and bats are introduced, the incorporation of these flight characteristics into the development of FWAVs is elucidated, and the utilization of these flight characteristics in the development of FWAVs is explained. Next, the classification and analysis of flapping mechanisms are conducted based on wing motion and the strategy for improving aerodynamic performance. Lastly, the current research gap is elucidated, and potential future directions for further research are proposed. This review can serve as a guide during the early development stage of FWAVs.