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Rivers in the Lake Victoria Basin support a multitude of ecosystem services, and the economies of the riparian countries (Kenya, Tanzania, Uganda, Rwanda, and Burundi) rely on their discharge, but projections of their future discharges under various climate change scenarios are not available. Here, we apply Vector Autoregressive Moving Average models with eXogenous variables (VARMAX) statistical models to project hydrological discharge for 23 river catchments for the 2015-2100 period, under three representative concentration pathways (RCPs), namely RCPs 2.6, 4.5, and 8.5. We show an intensification of future annual rainfall by 25% in the eastern and 5-10% in the western part of the basin. At higher emission scenarios, the October to December season receives more rainfall than the March to May season. Temperature projections show a substantial increase in the mean annual minimum temperature by 1.3-4.5 degrees C and warming in the colder season (June to September) by 1.7-2.9 degrees C under RCP 4.5 and 4.9 degrees C under RCP 8.5 by 2085. Variability in future river discharge ranges from 5-267%, increases with emission intensity, and is the highest in rivers in the southern and south eastern parts of the basin. The flow trajectories reveal no systematic trends but suggest marked inter-annual variation, primarily in the timing and magnitude of discharge peaks and lows. The projections imply the need for coordinated transboundary river management in the future.