Two hypotheses about primate cognition are proposed. First, it is proposed that primates, but not other mammals, understand categories of relations among external entities. In the physical domain primates have special skills in tasks such as oddity, transitivity, and relation matching that require facility with relational categories; in the social domain primates have special skills in understanding the third-party social relationships that hold among other individuals in their groups. Second, it is proposed that humans, but not other primates, understand the causal and intentional relations that hold among external entities. In the physical domain only humans understand causal forces as mediating the connection between sequentially ordered events; in the social domain only humans understand the behavior of others as intentionally directed and controlled by desired outcomes. Both these uniquely primate and these uniquely human cognitive skills are hypothesized to have their origins in adaptations for negotiating complex social interactions.