When the players in a game G can communicate with a referee via quantum technology (e.g. by sending emails composed on a quantum computer), their strategy sets naturally expand to include quantum superpositions of pure strategies. These superpositions lead to probability distributions among payoffs that would be impossible if players were restricted to classical mixed strategies. Thus the game G is replaced by a much larger “quantum game” GQ. When G is a 2 x 2 game, the strategy spaces of GQ are copies of the threedimensional sphere S3; therefore a mixed strategy is an arbitrary probability distribution on S3. These strategy spaces are so large that Nash equilibria can be difficult to compute or even to describe. The present paper largely overcomes this difficulty by classifying all mixed-strategy Nash equilibria in games of the form GQ. One result is that we can confine our attention to probability distributions supported on at most four points of S3; another is that these points must lie in one of several very restrictive geometric configurations. A stand-alone Appendix summarizes the relevant background from quantum mechanics and quantum game theory.