We introduce and study coverage games - a novel framework for multi-agent planning in settings in which a system operates several agents but does not have full control on them, or interacts with an environment that consists of several agents. The game is played between a coverer, who has a set of objectives, and a disruptor. The coverer operates several agents that interact with the adversarial disruptor. The coverer wins if every objective is satisfied by at least one agent. Otherwise, the disruptor wins. Coverage games thus extend traditional two-player games with multiple objectives by allowing a (possibly dynamic) decomposition of the objectives among the different agents. They have many applications, both in settings where the system is the coverer (e.g., multi-robot surveillance, coverage in multi-threaded systems) and settings where it is the disruptor (e.g., prevention of resource exhaustion, ensuring non-congestion). We first study the theoretical properties of coverage games, including determinacy, and the ability to a priori decompose the objectives among the agents. We then study the problems of deciding whether the coverer or the disruptor wins. Besides a comprehensive analysis of the tight complexity of the problems, we consider interesting special cases, such as the one-player cases and settings with a fixed number of agents or objectives.