The Hippocratic Oath has guided the conduct of physicians for centuries. Inspired by its tenet of preserving privacy, we argue that future database systems must include responsibility for the privacy of data they manage as a founding tenet. We enunciate the key privacy principles for such Hippocratic database systems. We propose a strawman design for Hippocratic databases, identifythe technical challenges and problems in designing such databases, and suggest some approaches that may lead to solutions. Our hope is that this paper will serve to catalyze a fruitful and exciting direction for future database research.