Natural selection could determine whether corals persist under expected climate change.

Marine heatwaves are intensifying under climate change, exposing populations of reef-building corals to mass mortality and intense selective pressure. It remains unknown whether adaptation can keep pace with warming and maintain reef functioning. We have developed an eco-evolutionary metapopulation model for , an ecologically important yet thermally sensitive coral taxon. We found that, although corals have some adaptation capacity, they will suffer severe heatwave-induced declines over the coming decades. For a future in which emissions lead to ~3°C of global warming, natural selection could allow populations to persist, albeit in severely depleted states with elevated extinction risk and potential loss of ecosystem functioning. Yet, for thermally sensitive coral populations to thrive beyond 2050, there must be rapid reductions of greenhouse gas emissions that limit global warming to 2°C.