Summary1. Interpretation of structural evolution in a group such as the Sarcopterygii requires consideration of a combination of all possible functions, rather than single functions.2. The Dipnoi are probably more closely related to the Crossopterygii than to other groups of fishes. The Sarcopterygii are a ‘natural’ group. Certain characters in common between the elasmobranchs and the Dipnoi or Coelacanthini seem to be the result of convergent evolution.3. Evolution of the skull, in connexion with both respiratory and feeding mechanisms, has resulted in extreme specialization in all Sarcopterygii. The crossopterygian intracranial kinesis has evolved from an earlier mobility between the skull and neck and is adapted for increasing the power of the bite and for enclosing the prey from both above and below, in addition to other factors. Adaptive radiation is seen in the feeding mechanisms of all forms. The evolution of the Amphibia proceeded through elongation of the anterior division of the skull (which is not correlated with any changes in brain morphology) and loss of the kinetic mechanism in this sequence is at least partially associated with improved buccal pumping mechanisms for lung ventilation.4. Adaptive radiation of the respiratory system in Dipnoi shows a progressive increase in the use of aerial respiration. The aquatic condition seen inNeoceratodusis probably secondary. Comparison of the three living genera shows a striking correlation between respiratory physiology and habit. There is little indication of reduction of the branchial respiratory system in known Rhipidistia, in which respiration was probably primarily aquatic. In Dipnoi and Rhipidistia, evolution of the lung allowed a partial control of the hydrostatic properties of the body. In coelacanths, aerial respiration was abandoned, except in certain secondarily freshwater forms, and the single lung is modified as an organ of hydrostatic balance. These changes are reflected in the over‐all body proportions.5. Locomotion in Sarcopterygii (except the coelacanthsLaugiaandPiveteauia) is adapted for contact with the substrate in relatively shallow water in most cases. Adaptive radiation of the locomotor apparatus is seen with respect to the relative roles and functions of the paired and unpaired fins, over‐all body shape, caudal fin shape, and absolute size. An important function of the pectoral fins in advanced Rhipidistia was in supporting the body in shallow water and thus aiding lung ventilation.6. Aestivation is an early feature of dipnoan biology, but was not evolved in Rhipidistia. The common faculty of urea production via the ornithine cycle and urea retention in coelacanths and dipnoans are adaptations to conditions in which the body tissues may become dehydrated (salt water and desiccation, respectively). The common pattern of nitrogen metabolism seems to have evolved during a marine phase in sarcopterygian evolution.7. There is evidence that the earliest members of all sarcopterygian lines included marine forms. However, the subsequent major radiations of Dipnoi and Rhipidistia occurred in fresh waters. The distribution of Sarcopterygii was entirely tropical. The late Palaeozoic distribution of the freshwater forms seems to offer evidence for the occurrence of Continental Drift. Coelacanths were primarily coastal fishes.8. The evolution of a major group of organisms requires a different pattern of evolutionary change than that by which adaptive radiations are produced. It evolves the structural and temporal correlation of modification in a number of different functional systems rather than the separate modification of each system without reference to other systems.9. The first tetrapods evolved in a highly seasonal swampy environment on the shores of inland lakes or rivers, in permanently moist con