This report discusses the topic of periodic cruise trajectories for hypersonic flight. An extensive review of previous work associated with periodic cruise trajectories for subsonic, supersonic and hypersonic flight is presented to provide the background for this investigation. The primary objective of this report is to discuss why periodic cruise trajectories lead to near fuel-optimal trajectories from a heuristic, mathematical and computational perspective with air breathing propulsion. Results to date indicate that periodic achieves greater fuel savings by exchanging kinetic and potential energy more efficiently. The vehicle attempts to chatter back and forth between where the vehicle wants to fly for optimum aerodynamic and propulsive performance. Results from computational simulations are inconclusive and require further work to define appropriate interfaces for aerodynamic and propulsion data decks for input into the POST software. The notional design of a vehicle to fly periodic hypersonic cruise trajectories was improved by including concepts for engine installation, flight controls and by including considerations for off-design performance. This notional design provides a better starting point for more serious and complete vehicle design studies.