Insects, birds, and bats that power and control flight by flapping their wings perform excellent flight stability and maneuverability by rapidly and continuously varying their wing motions. This article provides an overview of the state of the art of vortex-dominated, unsteady flapping aerodynamics from the viewpoint of diversity and uniformity associated with dominant vortices, particularly of the relevant physical aspects of the flight of insects and vertebrates in the low- and intermediate-Reynolds-number (Re) regime of 10⁰ to 10⁶. After briefly describing wing morphology and kinematics, we discuss the main vortices generated by flapping wings and the aerodynamic forces associated with these structures, focusing on leading-edge vortices (LEVs), wake vortices, and vortices generated by wing motions over a broad Re range. The LEVs are intensified by dynamic wing morphing in bird and bat flight, producing a significantly elevated vortex lift. The complex wake vortices are the footprints of lift generation; thus, the time-averaged vortex lift can be estimated from wake velocity data. Computational fluid dynamics modeling, quasi-steady models, and vortex lift models are useful tools to elucidate the intrinsic relationships between the lift and the dominant vortices in the near- and far-fields in flapping flight.