River avulsion occurs when a river channel suddenly switches course. Avulsing rivers create new pathways on the floodplain and the associated flooding can profoundly affect society, with records of these disasters stretching back millennia into human history. River avulsions are thought to occur when the water column becomes perched above the floodplain or when the slope down the flanks of the channel provides a steeper descent than the existing river channel. I tested these classical ideas by quantifying the topography around avulsing rivers and show that these mechanisms, historically invoked separately, work together. Near coasts, rivers avulse when the slope away from the channel is steeper, not because they are perched. The opposite is true near mountain fronts; on fans, the alternative paths are similarly steep to the downstream path, so rivers avulse when they are perched above the surrounding landscape. I reconcile these findings and present a new theoretical framework that identifies which rivers are vulnerable to avulsion and predicts the path of an avulsing river. These first-order rules of avulsion suggest that avulsion risks are underestimated in many coastal environments and that probabilistic predictions of avulsion pathfinding can efficiently map hazards with minimal information. Applying these principles for risk assessment could particularly benefit the Global South, which is disproportionately affected by avulsions.