Conventionally, the heaviest rainfall has been linked to the tallest, most intense convective storms. However, the global picture of the linkage between extreme rainfall and convection remains unclear. Here we analyse an 11-year record of spaceborne precipitation radar observations and establish that a relatively small fraction of extreme convective events produces extreme rainfall rates in any region of the tropics and subtropics. Robust differences between extreme rainfall and convective events are found in the rainfall characteristics and environmental conditions, irrespective of region; most extreme rainfall events are characterized by less intense convection with intense radar echoes not extending to extremely high altitudes. Rainfall characteristics and environmental conditions both indicate the importance of warm-rain processes in producing extreme rainfall rates. Our results demonstrate that, even in regions where severe convective storms are representative extreme weather events, the heaviest rainfall events are mostly associated with less intense convection.