The Mori cone is a fundamental, often elusive, invariant of an algebraic variety and is the central object of study in higher dimensional algebraic geometry. In this talk I will explain Fulton's conjecture, which predicts a very simple description of the Mori cone of the moduli space of curves. I'll show how one can naturally obtain upper and lower bounds for the Mori cone of a large class of varieties. In the case of the moduli space of curves, the upper bound is the cone described by Fulton's conjecture. In particular, this gives a new possibitlity for the Mori cone and a new perspective on Fulton's conjecture.