As AI systems quickly improve in both breadth and depth of performance, they lend themselves to creating increasingly powerful and realistic agents, including the possibility of agents modeled on specific people. We anticipate that within our lifetimes it may become common practice for people to create a custom AI agent to interact with loved ones and/or the broader world after death. We call these generative ghosts, since such agents will be capable of generating novel content rather than merely parroting content produced by their creator while living. In this paper, we first discuss the design space of potential implementations of generative ghosts. We then discuss the practical and ethical implications of generative ghosts, including potential positive and negative impacts on individuals and society. Based on these considerations, we lay out a research agenda for the AI and HCI research communities to empower people to create and interact with AI afterlives in a safe and beneficial manner.