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Small drops can bounce indefinitely on a bath of the same liquid if the container is oscillated vertically at a sufficiently high acceleration. Here we show that bouncing droplets can be made to ‘walk’ at constant horizontal velocity on the liquid surface by increasing this acceleration. This transition yields a new type of localized state^,,, with particle–wave duality: surface capillary waves emanate from a bouncing drop, which self-propels by interaction with its own wave and becomes a walker. When two walkers come close, they interact through their waves and this ‘collision’ may cause the two walkers to orbit around each other^,,.