This chapter reviews recent developments, with a particular focus on the cognitive abilities of non-human primates, and discusses the implications for the ongoing debate about the evolution of human language. Theories of language usually rely on: an inventory of lexical items, a mechanism to relate these to mental concepts, and rules to construct an infinite number of structures from them. The chapter investigates a number of natural communication systems of non-human primates to understand how external events trigger signaling behavior, how they link with underlying mental representations, and how multiple signals are used to construct more complex structures. Human language tends to take place in a state of high mutual awareness between speakers and receivers; it is grounded in mutual knowledge, beliefs, and assumptions. Humans may be unique in their social cognitive ability to take into account what others know and assume about the world before they address them.