Despite the uncertainty in future climate-change impacts, it is often assumed that humans would be able to adapt to any possible warming. Here we argue that heat stress imposes a robust upper limit to such adaptation. Peak heat stress, quantified by the wet-bulb temperature T(W), is surprisingly similar across diverse climates today. T(W) never exceeds 31 degrees C. Any exceedence of 35 degrees C for extended periods should induce hyperthermia in humans and other mammals, as dissipation of metabolic heat becomes impossible. While this never happens now, it would begin to occur with global-mean warming of about 7 degrees C, calling the habitability of some regions into question. With 11-12 degrees C warming, such regions would spread to encompass the majority of the human population as currently distributed. Eventual warmings of 12 degrees C are possible from fossil fuel burning. One implication is that recent estimates of the costs of unmitigated climate change are too low unless the range of possible warming can somehow be narrowed. Heat stress also may help explain trends in the mammalian fossil record.