It has long been suggested that climate shapes land surface topography through interactions between rainfall, runoff and erosion in drainage basins1-4. The longitudinal profile of a river (elevation versus distance downstream) is a key morphological attribute that reflects the history of drainage basin evolution, so its form should be diagnostic of the regional expression of climate and its interaction with the land surface5-9. However, both detecting climatic signatures in longitudinal profiles and deciphering the climatic mechanisms of their development have been challenging, owing to the lack of relevant global data and to the variable effects of tectonics, lithology, land surface properties and human activities10,11. Here we present a global dataset of 333,502 river longitudinal profiles, and use it to explore differences in overall profile shape (concavity) across climate zones. We show that river profiles are systematically straighter with increasing aridity. Through simple numerical modelling, we demonstrate that these global patterns in longitudinal profile shape can be explained by hydrological controls that reflect rainfall-runoff regimes in different climate zones. The most important of these is the downstream rate of change in streamflow, independent of the area of the drainage basin. Our results illustrate that river topography expresses a signature of aridity, suggesting that climate is a first-order control on the evolution of the drainage basin.