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.