The evolution of speech is considered to be one of the hardest problems in science. Studies of the communicative abilities of our closest living relatives, the nonhuman primates, aim to contribute to a better understanding of the emergence of this uniquely human capability. Following a brief introduction over the key building blocks that make up the human speech faculty, I will focus on the question of meaning in nonhuman primate vocalizations. While nonhuman primate calls may be highly context specific, thus giving rise to the notion of ‘referentiality’, comparisons across closely related species suggest that this specificity is evolved rather than learned. Yet, as in humans, the structure of calls varies with arousal and affective state, and there is some evidence for effects of sensory-motor integration in vocal production. Thus, the vocal production of nonhuman primates bears little resemblance to the symbolic and combinatorial features of human speech, while basic production mechanisms are shared. Listeners, in contrast, are able learning the meaning of new sounds. A recent study using artificial predator shows that this learning may be extremely rapid. Furthermore, listeners are able to integrate information from multiple sources to make adaptive decisions, which renders the vocal communication system as a whole relatively flexible and powerful. In conclusion, constraints at the side of vocal production, including limits in social cognition and motivation to share experiences, rather than constraints at the side of the recipient explain the differences in communicative abilities between humans and other animals.