Semantic communication is about transmitting mental representations of reality. Three research questions address the nature of this process in primates. Can primates produce signals that are meaningful in a lexical sense? Are they capable of compositional semantics? Can they create and infer meaning by integrating context and intention? There is good evidence that, as recipients, primates have capacities at all three levels, whereas for signallers the evidence is less compelling. This difference may have cognitive roots, due to the fact that primate signallers are typically engaged in the here-and-now and, unlike humans, less able to refer to memory content. Future research will have to clarify what mental structures primates can take into account during communication, including entities that are not physically present.