This paper defends a gestural origins hypothesis about the evolution of enhanced communication and language in the hominin lineage. The paper shows that we can develop an incremental model of language evolution on that hypothesis, but not if we suppose that language originated in an expansion of great ape vocalization. On the basis of the gestural origins hypothesis, the paper then advances solutions to four classic problems about the evolution of language: (i) why did language evolve only in the hominin lineage? (ii) why is language use an evolutionarily stable form of informational cooperation, despite the fact that hominins have diverging evolutionary interests? (iii) how did stimulus independent symbols emerge? (iv) what were the origins of complex, syntactically organized symbols? The paper concludes by confronting two challenges: those of testability and of explaining the gesture-to-speech transition; crucial issues for any gestural origins hypothesis.