How Do Spillways Work?

We normally build a dam to hold water back and store it for use in water supply, irrigation, hydropower, or flood control. But sometimes we have to let some water go. Whether we need it downstream or the impounded water behind the dam is simply too full to store any more, nearly every dam needs a spillway to safely discharge water. The spillway is a critical part of any dam and often the most complex component. So how does it work?