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Groundwater in sub-Saharan Africa supports livelihoods and poverty alleviation(1,2), maintains vital ecosystems, and strongly influences terrestrial water and energy budgets(3). Yet the hydrological processes that govern groundwater recharge and sustainability-and their sensitivity to climatic variability-are poorly constrained(4,5). Given the absence of firm observational constraints, it remains to be seen whether model-based projections of decreased water resources in dry parts of the region(4) are justified. Here we show, through analysis of multidecadal groundwater hydrographs across sub-Saharan Africa, that levels of aridity dictate the predominant recharge processes, whereas local hydrogeology influences the type and sensitivity of precipitation-recharge relationships. Recharge in some humid locations varies by as little as five per cent (by coefficient of variation) across a wide range of annual precipitation values. Other regions, by contrast, show roughly linear precipitation-recharge relationships, with precipitation thresholds (of roughly ten millimetres or less per day) governing the initiation of recharge. These thresholds tend to rise as aridity increases, and recharge in drylands is more episodic and increasingly dominated by focused recharge through losses from ephemeral overland flows. Extreme annual recharge is commonly associated with intense rainfall and flooding events, themselves often driven by large-scale climate controls. Intense precipitation, even during years of lower overall precipitation, produces some of the largest years of recharge in some dry subtropical locations. Our results therefore challenge the 'high certainty' consensus regarding decreasing water resources(4) in such regions of sub-Saharan Africa. The potential resilience of groundwater to climate variability in many areas that is revealed by these precipitation-recharge relationships is essential for informing reliable predictions of climate-change impacts and adaptation strategies.