Warming of aquatic environments as a result of climate change is already having measurable impacts on fishes, manifested as changes in phenology, range shifts and reductions in body size. Understanding the physiological mechanisms underlying these seemingly universal patterns is crucial if we are to reliably predict the fate of fish populations with future warming. This includes an understanding of mechanisms for acute thermal tolerance, as extreme heatwaves may be a major driver of observed effects. The hypothesis of gill oxygen limitation (GOL) is claimed to explain asymptotic fish growth, and why some fish species are decreasing in size with warming; but its underlying assumptions conflict with established knowledge and direct mechanistic evidence is lacking. The hypothesis of oxygen- and capacity-limited thermal tolerance (OCLTT) has stimulated a wave of research into the role of oxygen supply capacity and thermal performance curves for aerobic scope, but results vary greatly between species, indicating that it is unlikely to be a universal mechanism. As thermal performance curves remain important for incorporating physiological tolerance into models, we discuss potentially fruitful alternatives to aerobic scope, notably specific dynamic action and growth rate. We consider the limitations of estimating acute thermal tolerance by a single rapid measure whose mechanism of action is not known. We emphasise the continued importance of experimental physiology, particularly in advancing our understanding of underlying mechanisms, but also the challenge of making this knowledge relevant to the more complex reality.