This paper presents a generalization of the satisfaction equilibrium (SE) for games in satisfaction form (SF), namely, generalized satisfaction equilibrium (GSE). In games in SF, players choose their actions aiming to satisfy a constraint that depends on the actions of all the others. Given an action profile, players satisfying their individual constraint are said to be satisfied, whereas the others, unsatisfied. At a GSE, players that are unsatisfied cannot unilaterally deviate to satisfy their individual constraint. This game formulation, as well as the GSE, are particularly adapted to model problems of service-level provisioning in communications networks. The existence of at least one GSE in mixed strategies is proven for certain classes of game in SF. The pure-strategy GSE problem is closely related to the constraint satisfaction problem and finding a pure-strategy GSE with a given number of satisfied agents is proved to be NP-hard. For certain games in SF, it is shown that the satisfaction response dynamics converges to a GSE. Finally, Bayesian games in SF, and the corresponding GSE, are introduced. Finally, a series of examples in wireless communications in which GSE plays a more relevant role than other equilibrium concepts, e.g., generalized Nash equilibrium and satisfaction equilibrium, are discussed.