Extreme precipitation is a considerable contributor to meteorological disasters and there is a great need to mitigate its socioeconomic effects through skilful nowcasting that has high resolution, long lead times and local details. Current methods are subject to blur, dissipation, intensity or location errors, with physics-based numerical methods struggling to capture pivotal chaotic dynamics such as convective initiation and data-driven learning methods failing to obey intrinsic physical laws such as advective conservation. We present NowcastNet, a nonlinear nowcasting model for extreme precipitation that unifies physical-evolution schemes and conditional-learning methods into a neural-network framework with end-to-end forecast error optimization. On the basis of radar observations from the USA and China, our model produces physically plausible precipitation nowcasts with sharp multiscale patterns over regions of 2,048 km × 2,048 km and with lead times of up to 3 h. In a systematic evaluation by 62 professional meteorologists from across China, our model ranks first in 71% of cases against the leading methods. NowcastNet provides skilful forecasts at light-to-heavy rain rates, particularly for extreme-precipitation events accompanied by advective or convective processes that were previously considered intractable.