I will discuss the basic theory of Koszul modules, which were originally introduced by Papadima and Suciu as a tool to study topological invariants of groups. A special instance of Koszul modules had previously appeared in David Eisenbud's "Green's conjecture: an orientation for algebraists", where he proposed several programs for proving the Green conjecture for generic canonical curves. I will explain a vanishing theorem for Koszul modules that completes one of these programs, providing an alternative approach to the original proof of Voisin for the generic Green conjecture. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.