Event structures form a canonical model of concurrent behaviour which has a natural game-theoretic interpretation. This game-based interpretation was initially given for zero-sum concurrent games. This paper studies an extension of such games on event structures to include a much wider class of game types and solution concepts. The extension permits modelling scenarios where, for instance, cooperation or independent goal-driven behaviour of computer agents is desired. Specifically, we will define non-zero-sum games on event structures, and give full characterisations---existence and completeness results---of the kinds of games, payoff sets, and strategies for which Nash equilibria and subgame perfect Nash equilibria always exist. The game semantics of various logics and systems are outlined to illustrate the power of this framework.