Because anthropogenic land fragmentation will prevent large terrestrial mammals from dispersing, the future welfare of these mammals will depend on their abilities to cope with changing climates in their existing habitats, and the capacity of phenotypic plasticity to extend those abilities. Correlative and mechanistic models have been constructed to predict the welfare of mammals under climate change. Mechanistic models, currently preferred, require an appreciation of thermal physiology, sometimes wanting. For example, some mechanistic models assume that any conditions outside the thermoneutral zone (TNZ) will be deleterious, but there is no evidence that TNZ has any relevance to mammalian ability to cope with climate change. Desert antelope live and reproduce at air temperatures well above their TNZ. TNZ usually is defined by air temperature and the heat loads experienced by mammals depend on much more than air temperature, with direct, reflected and reradiated solar radiation often forming the major component of the load. Some models, whether based on TNZ or not, ignore radiation. Some mammalian models inappropriately import concepts from ectothermy. The concept of temperature-dependent optimal performance may provide insight into ecology of ectotherms, but is less relevant to mammals. Homeothermy is not a requirement for sustained existence in mammals. Large mammals under heat load maintain homeothermy if they are resourced well, but relax homeothermy if water or food become limited. Their temperatures are unpredictable without knowledge of their nutrition and hydration. Large mammals give priority to water regulation over thermoregulation. If water is available, large mammals, including elephants that neither sweat nor pant, use evaporative cooling to support homeothermy. The use of evaporative cooling is not an unsustainable state, as is assumed by models that employ upper limits of TNZs to predict lethal limits. Another concept imported inappropriately from ectotherm thermal physiology is the assumption that skin temperature is equal to air temperature. Our measurements, including infrared thermography, have confirmed that the skin temperature of free-living large mammals may be far from air temperature. The mechanistic models also tend to ignore the wealth of information available from humans. For example, studies of human acclimation show that acclimatization to heat lowers body core temperature; some models assume that mammalian core temperature is insensitive to environment. Contrary to some models that predict the opposite, we have shown that core temperature decreases with increasing body mass in large mammals. A widely-acknowledged limitation of both correlative and mechanistic models is that they tend to address the welfare of species independently of their biotic environments. We and others have shown that the future welfare of large mammals will depend not only on their abilities but also on the resilience of their plant and animal food under climate change. Finally, all models suffer from uncertainty about how mammals will respond physiologically to climate change. Better predictions will require long-term studies, requiring biologging, of the thermal physiology of large mammals living free in their natural habitats and unstressed by human observers, studies that we have begun.