Investigating in depth the mechanisms underlying human and non-human primate intentional communication systems (involving gestures, vocalisations, facial expressions and eye behaviours) can shed light on the evolutionary roots of language. Reports on non-human primates, particularly great apes, suggest that gestural communication would have been a crucial prerequisite for the emergence of language, mainly based on the evidence of large communication repertoires and their associated multifaceted nature of intentionality that are key properties of language. Such research fuels important debates on the origins of gestures and language. We review here three non-mutually exclusive processes that can explain mainly great apes' gestural acquisition and development: phylogenetic ritualisation, ontogenetic ritualisation, and learning via social negotiation. We hypothesise the following scenario for the evolutionary origins of gestures: gestures would have appeared gradually through evolution via signal ritualisation following the principle of derived activities, with the key involvement of emotional expression and processing. The increasing level of complexity of socioecological lifestyles and associated daily manipulative activities might then have enabled the acquisition and development of different interactional strategies throughout the life cycle. Many studies support a multimodal origin of language. However, we stress that the origins of language are not only multimodal, but more broadly multicausal. We propose a multicausal theory of language origins which better explains current findings. It postulates that primates' communicative signalling is a complex trait continually shaped by a cost-benefit trade-off of signal production and processing of interactants in relation to four closely interlinked categories of evolutionary and life cycle factors: species, individual and context-related characteristics as well as behaviour and its characteristics. We conclude by suggesting directions for future research to improve our understanding of the evolutionary roots of gestures and language.