Nonhuman primates mostly communicate not only with a rich vocal repertoire but also with manual and body gestures. In contrast to great apes, this latter communicative gestural system has been poorly investigated in monkeys. In the last 15 years, the gestural research we conducted in the baboons Papio anubis, an Old World monkey species, have shown potential direct evolutionary continuities with some key properties of language such as intentionality, referentiality, learning flexibility as well as its underlying lateralization and hemispheric specialization of the brain. According to these collective findings, which are congruent with the ones reported in great apes, it is thus not excluded that features of gestural communication shared between humans, great apes and baboons, may have played a critical role in the phylogenetic roots of language and dated back, not to the Hominidae evolution, but rather to their much older Catarrhine common ancestor 25-40 million years ago.