Abstract: A computational theory of primate social intelligence is proposed in which primates represent social situations internally by discrete symbol structures, called scripts. Three well-defined computational operations on scripts are sufficient to support social learning, planning, and prediction. This gives a formal, predictive model with which to analyse how primate social knowledge is acquired, as well as how it is used. The theory is compared with primate data, such as Cheney and Seyfarth's observations of vervet monkeys. It gives simple, understandable script-based analyses of many observed phenomena--such as the recognition and use of kin relations, learning of alarm calls, habituation to calls, knowledge of rank, tactical deception, and attachment behaviour. I argue that a tight, concise theory of social cognition, such as script theory, is needed to explain the rapid learning and social guile seen in primates. It also has the benefits of simplicity and testability. The extension of scripts to incorporate a primate theory of mind is described in a subsequent paper.