We introduce and study Minkowski games. These are two-player games, where the players take turns to choose positions in R<sup<d</sup< based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded, and the other wants to escape to infinity; as well as safety games, where one player wants to stay within a prescribed set, while the other wants to leave it. We provide some general characterizations of which player can win such games and explore the computational complexity of the associated decision problems. A natural representation of boundedness games yields coNP-completeness, whereas the safety games are undecidable.