Comparative research on the origins of human language often focuses on a limited number of language-related cognitive functions or anatomical structures that are compared across species. The underlying assumption of this approach is that a single or a limited number of factors may crucially explain how language appeared in the human lineage. Another potentially fruitful approach is to consider human language as the result of a (unique) assemblage of multiple cognitive and anatomical components, some of which are present in other species. This paper is a first step in that direction. It focuses on the baboon, a non-human primate that has been studied extensively for years, including several brain, anatomical, cognitive and cultural dimensions that are involved in human language. This paper presents recent data collected on baboons regarding (1) a selection of domain-general cognitive functions that are core functions for language, (2) vocal production, (3) gestural production and cerebral lateralization, and (4) cumulative culture. In all these domains, it shows that the baboons share with humans many cognitive or brain mechanisms which are central for language. Because of the multidimensionality of the knowledge accumulated on the baboon, that species is an excellent nonhuman primate model for the study of the evolutionary origins of language.