  We discuss recent progress on syzygies of curves, including proofs of Green's
and Gonality Conjectures as well as applications of Koszul cycles to the study
of the birational geometry of various moduli spaces of curves. We prove a
number of new results, including a complete solution to Green's Conjecture for
arbitrary hexagonal curves. Finally, we propose several new conjectures on
syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as
well as a strong Maximal Rank Conjecture for generic curves. To appear in the
Proceedings of Clay Mathematical Institute.
