Our understanding of the aerodynamics of flapping animal flight is largely based on the quasi-steady assumption: the instantaneous aerodynamic forces on a flapping wing are assumed to be identical with those which the wing would experience in steady motion at the same instantaneous speed and angle of attack. Research up to a decade ago showed that the assumption was sufficient to explain the flight of the vast majority of animals, but did not rule out the possibility that alternative aerodynamic mechanisms were employed instead. Results are presented here for four hovering animals for which the quasi-steady explanation fails. These animals apparently use lift mechanisms that rely on vortices shed during the rotational motion of the wing at either end of the wingbeat. The postulated rotational lift mechanisms should also apply to other hovering animals, even though the quasi-steady assumption could explain their flight. Measurements of the wing forces produced by locusts cast doubt on the validity of the quasi-steady assumption for fast forward flight as well.