We introduce the game-theoretical model of budget games and analyze the existence of pure Nash equilibria. In a budget game, players compete over resources with a limited budget. As his strategy, each player decides between a finite number of demand vectors. Each demand vector contains one non-negative demand for every resource. Provided the total demand by all players on a single resource does not exceed its budget, the utility each player receives from that resource equals his demand. Otherwise, the budget is split proportionally to the demands. For any combination of player and resource, the corresponding demand is directly tied to the players strategy and can change during the best-response dynamic. After showing that pure Nash equilibria generally do not exist in budget games, we consider several alternative concepts. (1) Ordered budget games are a variation which emphasizes the order in which the players choose their strategies. They are exact potential games for which even the existence of super-strong pure Nash equilibria can be guaranteed and strong pure Nash equilibria can be computed efficiently. (2) In an alpha-approximate pure Nash equilibrium, no unilateral strategy change increases the utility of the corresponding player by more than some constant factor alpha. We give upper and lower bound on alpha such that approximate pure Nash equilibria are guaranteed. In addition, we look at an approximate version of the best-response dynamic which converges quickly and can also be used to compute a strategy profile which approximates the maximal social welfare. (3) By restricting the structure of the strategy spaces, we restore pure Nash equilibria to budget games. We focus on singleton and matroid budget games. We also argue that computing the socially optimal solution is equivalent for both budget games and ordered budget games and NP-hard in both cases.