Negative societal impacts can result from intense individual downpours, the accumulation of rainfall over a day or more, or a combination of these. Accumulation is reasonably well captured by daily reporting rain gauges, but rainfall intensity is not. Ten years of data from the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR) are used to describe the spatial and seasonal distributions of instantaneous rainfall intensity with an emphasis on how these differ from the distributions of mean daily accumulation. Over tropical land, the rainy season, when rainfall is most frequent, does not coincide with the highest mean intensity. Rather, intensity peaks just before the rainy season. This offset is most obvious in the pre-onset and post-onset months in monsoon regions and it is also evident in equatorial regions without a well-defined dry and rainy season. Most seasonal variations in rainfall intensity can be explained as parallel variations in the occurrence of convective, relative to stratiform, precipitation. However, regional differences in rainfall intensity are related to differences in the intensity of convection itself. Compared with seasonal changes in intensity over land, variations in convective precipitation fraction over tropical oceans are trivial, and the modest seasonal changes in the intensity of rainfall parallel those of frequency. These findings suggest that studies of precipitation extremes under global warming should (1) explicitly tackle the question of changes in the intensity of rainfall separately from changes in daily rainfall accumulation and (2) consider the different qualities of extreme precipitation events over ocean and over land.