A broad sheet of water flowing over non-cohesive sediment typically breaks up into a network of interconnected channels called a braided stream (Fig. 1). The dynamics of such networks are complex; channels migrate laterally, split, rejoin and develop bars, with the flow shifting unpredictably from one part of the network to another. Many processes are known to operate in a braided river1-3, but it is unclear which of these are essential to explain the observed dynamics. We describe here a simple, deterministic numerical model of water flow over a cohesionless bed that captures the main spatial and temporal features of real braided rivers. The patterns arise from local scour and deposition caused by a nonlinear dependence of bedload sediment flux on water discharge. Although the morphology of the resulting network depends in detail on the sediment-transport rule used in the model, our results suggest that the only factors essential for braiding are bedload sediment transport and laterally unconstrained free-surface flow.