Pterosaurs have fascinated scientists and nonscientists alike for over 200 years, as one of the three known clades of vertebrates to have evolved flapping flight. The smallest pterosaurs were comparable in size to the smallest extant birds and bats, but the largest pterosaurs were vastly larger than any extant flier. This immense size range, coupled with poor preservation and adaptations for flight unknown in extant vertebrates, have made interpretations of pterosaur flight problematic and often contentious. Here we review the anatomical, evolutionary, and phylogenetic history of pterosaurs, as well as the views, perspectives, and biases regarding their interpretation. In recent years, three areas of pterosaur biology have faced challenges and made advances: structure of the wing membrane, function of the pteroid, body size and mass estimates, as well as flight mechanics and aerodynamics. Comparative anatomical and fossil study, simulated bone loading, and aerodynamic modeling have all proved successful in furthering our understanding of pterosaur flight. We agree with previous authors that pterosaurs should be studied as pterosaurs, a diverse but phylogenetically, anatomically, and mechanically constrained clade that can offer new insights into the diversity of vertebrate flight.