Research in anthropology has shown that kin terminologies have a complex combinatorial structure and vary systematically across cultures. This article argues that universals and variation in kin terminology result from the interaction of (1) an innate conceptual structure of kinship, homologous with conceptual structure in other domains, and (2) principles of optimal, “grammatical” communication active in language in general. Kin terms from two languages, English and Seneca, show how terminologies that look very different on the surface may result from variation in the rankings of a universal set of constraints. Constraints on kin terms form a system: some are concerned with absolute features of kin (sex), others with the position (distance and direction) of kin in “kinship space,” others with groups and group boundaries (matrilines, patrilines, generations, etc.). Also, kin terms sometimes extend indefinitely via recursion, and recursion in kin terminology has parallels with recursion in other areas of language. Thus the study of kinship sheds light on two areas of cognition, and their phylogeny. The conceptual structure of kinship seems to borrow its organization from the conceptual structure of space, while being specialized for representing genealogy. And the grammar of kinship looks like the product of an evolved grammar faculty, opportunistically active across traditional domains of semantics, syntax, and phonology. Grammar is best understood as an offshoot of a uniquely human capacity for playing coordination games.