Humans are the only species that can speak. Nonhuman primates, however, share some "domain-general" cognitive properties that are essential to language processes. Whether these shared cognitive properties of humans and nonhuman primates are the result of a continuous or convergent evolution can be investigated by comparing their respective underlying structure: the brain. Key areas associated with language processes are the Planum Temporale, Broca's Area, the Arcuate Fasciculus, Cingulate Sulcus, The Insula, Superior Temporal Sulcus, the Inferior Parietal lobe and the Central Sulcus.These structures share a fundamental feature: They are functionally and also structurally specialised to one hemisphere. Interestingly, several nonhuman primate species, such as chimpanzees and baboons, show human-like structural brain asymmetries for areas homologous to these key-markers of functional language lateralisation. The question arises, then, for what function did these asymmetries arise in non-linguistic primates, if not for language per se? In an attempt to provide some answers, we review the literature on the lateralisation of the gestural communication system, which may represent the missing behavioural link to brain asymmetries for language area's homologues in our common ancestor.