Linguists interested in language evolution tend to focus on combinatorial features and rightly point out the lack of comparable evidence in animal communication. However, human language is based on various unique capacities, such as a motor capacity of sophisticated vocal control and a cognitive capacity of acting on others' psychological states. These features are only present in very rudimentary forms in non-human primates, suggesting they have evolved more recently in the human lineage. Here, the evidence from recent fieldwork for precursors of these abilities is reviewed, notably sequence-based semantic communication, vocal tract control, and audience awareness. Overall, there is evidence for both continuity and discontinuity when comparing modern primate and human communication, suggesting that the origin of language is the result of multiple gradual transitions from earlier forms of primate-like communication and social cognition, rather than a sudden and fundamental redesign in ancestral human communication and cognition.