Evolution has produced an astonishing array of organisms, but does it have limits and, if so, how are these overcome and how have they changed over the course of time? Here, I review models for describing and explaining existing diversity, and then explore parts of the evolutionary tree that remain empty. In an analysis of 32 forbidden states among eukaryotes, identified in major clades and in the three great habitat realms of water, land and air, I argue that no phenotypic constraint is absolute, that most constraints reflect a limited time–energy budget available to individual organisms, that natural selection is ultimately responsible for both imposing and overcoming constraints, including those normally ascribed to developmental patterns of construction and phylogenetic conservatism, and that increases in adaptive versatility in major clades together with accompanying new ecological opportunities have eliminated many constraints. Phenotypes that were inaccessible during the Early Palaeozoic era have evolved during later periods while very few adaptive states have disappeared. The filling of phenotypic space has proceeded cumulatively in three overlapping phases characterized by diversification at the biochemical, morphological and cultural levels.