Incomplete information, local interaction and random matching games all share a common mathematical structure. A type or player interacts with various subsets of the set of all types/players. A type/player's total payoff is additive in the payoffs from these various interactions. This paper describes a general class of interaction games and shows how each of these three classes of games can be understood as special cases. Techniques and results from the incomplete information literature are translated into this more general framework. A companion paper, Morris [1997], uses these techniques to derive new results concerning contagion in local interaction games.