Differential game logic (dGL) is a logic for specifying and verifying properties of hybrid games, i.e. games that combine discrete, continuous, and adversarial dynamics. Unlike hybrid systems, hybrid games allow choices in the system dynamics to be resolved adversarially by different players with different objectives. The logic dGL can be used to study the existence of winning strategies for such hybrid games, i.e. ways of resolving the player's choices in some way so that he wins by achieving

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... is objective for all choices of the opponent. Hybrid games are determined, i.e. from each state, one player has a winning strategy, yet computing their winning regions may take transfinitely many steps. The logic dGL, nevertheless, has a sound and complete axiomatization relative to any expressive logic. Separating axioms are identified that distinguish hybrid games from hybrid systems. Finally, dGL is proved to be strictly more expressive than the corresponding logic of hybrid systems by characterizing the expressiveness of both.