Abstract: There has been a long standing unproven conjecture by Fulton regarding the structure of the cone of curves of the moduli space of stable pointed genus zero curves.  This conjecture implies a similar conjecture for Hassett's weighted moduli spaces. In this talk I will review what is known and then discuss a proof of this conjecture in the special case of small weights.  The main technique involves constructing contractions of the moduli space to variations of a certain GIT quotient, using the universal properties of the moduli spaces.  Fulton's conjecture gives us an upper bound for the cone of curves.  By creating enough contractions we get a lower bound, which will turn out to agree with the upper bound.