Flight-defined as the ability to produce useful aerodynamic forces by flapping the wings-is one of the most striking adaptations in vertebrates. Its origin has been surrounded by considerable controversy, due in part to terminological inconsistencies, in part to phylogenetic uncertainty over the sister groups and relationships of birds, bats and pterosaurs, and in part to disagreement over the interpretation of the available fossil evidence and over the relative importance of morphological, mechanical and ecological specializations. Study of the correlation between functional morphology and mechanics in contemporary birds and bats, and in particular of the aerodynamics of flapping wings, clarifies the mechanical changes needed in the course of the evolution of flight. This strongly favours a gliding origin of tetrapod flight, and on mechanical and ecological grounds the alternative cursorial and fluttering hypotheses (neither of which is at present well-defined) may be discounted. The argumrnt is particularly strong in bats, but weaker in birds owing to apparent inconsistencies with the fossil evidence. However, study of the fossils of the Jurassic theropod dinosaur Archaeopteryx, the sister-group of the stem-group proto-birds, supports this view. Its morphology indicates adaptation for flapping flight at the moderately high speeds which would be associated with gliding, but not for the slow speeds which would be required for incipient flight in a running cursor, where the wingbeat is aerodynamically and kinematically considerably more complex. Slow flight in birds and bats is a more derived condition, and vertebrate flapping flight apparently evolved through a gliding stage.