One of the defining problems in the theory of algebraic curves in the last decades has been Green's Conjecture predicting that one can read the intrinsic geometry of a curve from the equations of its canonical embedding. I will describe how moduli spaces of pointed curves can be used in a rather surprising way to prove two statements intimately related to Green's Conjecture: the Minimal Resolution Conjecture linking the geometry of a canonical curve to the resolution of general subsets of its points and a conjecture of R. Lazarsfeld about the theta divisors of the powers of the normal bundle of a curve in its Jacobian.