\“Space drives,\” \“Warp drives,\” and \“Wormholes: \” these concepts may sound like science fiction, but they are being written about in reputable journals. To assess the implications of these emerging prospects for future spaceflight, NASA supported the Breakthrough Propulsion Physics Project from 1996 through 2002. This Project has three grand challenges: (1) Discover propulsion that eliminates the need for propellant; (2) Discover methods to achieve hyper-fast travel; and (3) Discover breakthrough methods to power spacecraft. Because these challenges are presumably far from fruition, and perhaps even impossible, a special emphasis is placed on selecting incremental and affordable research that addresses the critical issues behind these challenges. Of 16 incremental research tasks completed by the project and from other sponsors, about a third were found not to be viable, a quarter have clear opportunities for sequels, and the rest remain unresolved.