Bounding flight is a flight mode consisting of repeated cycles with alternating phases involving flapping and non-flapping. The bounding flight mode is treated as a periodic optimal control problem using an efficient optimization procedure to construct solutions. It is shown that bounding flight yields performance enhancements in the energy expenditure required for flying. In zero wind, bounding flight is superior to steady-state cruise wind at higher speeds. When flying against a headwind, the superiority of bounding flight is enlarged. It is shown that bounding flight requires less energy per range for a greater speed range. Particularly, the minimum energy per range of bounding flight is smaller than that possible with continuous flapping flight.