The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. It predicts that on a curve embedded by a line bundle of sufficiently high degree, the existence of a p-th syzygy is equivalent to the existence of a certain secant to the curve. I will discuss the history of this problem, then establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial cases, that is, when the line bundles that satisfy the corresponding secant condition form a divisor in the Jacobian of the curve.