The Archosauria are a highly successful group of vertebrates, and their evolution is marked by the appearance of diverse respiratory and metabolic strategies. This review examines respiratory function in living and fossil archosaurs, focusing on the anatomy and biomechanics of the respiratory system, and their physiological consequences. The first archosaurs shared a heterogeneously partitioned parabronchial lung with unidirectional air flow; from this common ancestral lung morphology, we trace the diverging respiratory designs of bird- and crocodilian-line archosaurs. We review the latest evidence of osteological correlates for lung structure and the presence and distribution of accessory air sacs, with a focus on the evolution of the avian lung-air sac system and the functional separation of gas exchange and ventilation. In addition, we discuss the evolution of ventilation mechanics across archosaurs, citing new biomechanical data from extant taxa and how this informs our reconstructions of fossils. This improved understanding of respiratory form and function should help to reconstruct key physiological parameters in fossil taxa. We highlight key events in archosaur evolution where respiratory physiology likely played a major role, such as their radiation at a time of relative hypoxia following the Permo-Triassic mass extinction, and their evolution of elevated metabolic rates. This article is part of the theme issue 'Vertebrate palaeophysiology'.