The Permian and Triassic were key time intervals in the history of life on Earth. Both periods are marked by a series of biotic crises including the most catastrophic of such events, the end-Permian mass extinction, which eventually led to a major turnover from typical Palaeozoic faunas and floras to those that are emblematic for the Mesozoic and Cenozoic. Here we review patterns in Permian-Triassic bony fishes, a group whose evolutionary dynamics are understudied. Based on data from primary literature, we analyse changes in their taxonomic diversity and body size (as a proxy for trophic position) and explore their response to Permian-Triassic events. Diversity and body size are investigated separately for different groups of Osteichthyes (Dipnoi, Actinistia, 'Palaeopterygii', 'Subholostei', Holostei, Teleosteomorpha), within the marine and freshwater realms and on a global scale (total diversity) as well as across palaeolatitudinal belts. Diversity is also measured for different palaeogeographical provinces. Our results suggest a general trend from low osteichthyan diversity in the Permian to higher levels in the Triassic. Diversity dynamics in the Permian are marked by a decline in freshwater taxa during the Cisuralian. An extinction event during the end-Guadalupian crisis is not evident from our data, but 'palaeopterygians' experienced a significant body size increase across the Guadalupian-Lopingian boundary and these fishes upheld their position as large, top predators from the Late Permian to the Late Triassic. Elevated turnover rates are documented at the Permian-Triassic boundary, and two distinct diversification events are noted in the wake of this biotic crisis, a first one during the Early Triassic (dipnoans, actinistians, 'palaeopterygians', 'subholosteans') and a second one during the Middle Triassic ('subholosteans', neopterygians). The origination of new, small taxa predominantly among these groups during the Middle Triassic event caused a significant reduction in osteichthyan body size. Neopterygii, the clade that encompasses the vast majority of extant fishes, underwent another diversification phase in the Late Triassic. The Triassic radiation of Osteichthyes, predominantly of Actinopterygii, which only occurred after severe extinctions among Chondrichthyes during the Middle-Late Permian, resulted in a profound change within global fish communities, from chondrichthyan-rich faunas of the Permo-Carboniferous to typical Mesozoic and Cenozoic associations dominated by actinopterygians. This turnover was not sudden but followed a stepwise pattern, with leaps during extinction events.
