How does a bike chain stay on the sprocket?

A bike chain stays on the sprocket because the teeth of the sprocket and the links of the chain are shaped in a way that allows the chain to pass over the sprocket smoothly without slipping off. The teeth of the sprocket are slightly inclined so that they gently guide the chain into the correct position, while the shape of the chain links prevents them from slipping off the teeth. The chain is also kept in place by the tension of the derailleur, which pulls it taut and prevents it from becoming too loose.