Some things can happen in our Universe, and others cannot. The laws of physics establish the boundary between possibility and impossibility. Physicists naturally spend most of their time thinking about the possible. In this lecture, however, we will make a brief reconnaissance across the frontier to study impossible things and discover the surprising connections between them. We will encounter standard science-fiction devices like time machines and faster-than-light spaceships -- as well as other, less-familiar prodigies including quantum cloners and bounded electromagnetic miracles. A safe return to the real world is unconditionally guaranteed.