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Short-duration (1–3 h) rainfall extremes can cause serious damage to societies through rapidly developing (flash) flooding and are determined by complex, multifaceted processes that are altering as Earth’s climate warms. In this Review, we examine evidence from observational, theoretical and modelling studies for the intensification of these rainfall extremes, the drivers and the impact on flash flooding. Both short-duration and long-duration (>1 day) rainfall extremes are intensifying with warming at a rate consistent with the increase in atmospheric moisture (~7% K⁻¹), while in some regions, increases in short-duration extreme rainfall intensities are stronger than expected from moisture increases alone. These stronger local increases are related to feedbacks in convective clouds, but their exact role is uncertain because of the very small scales involved. Future extreme rainfall intensification is also modulated by …