In this talk, I will discuss the relationship bewtween various projective compactifications of the moduli space of cubic threefolds; that is smooth degree three hypersurfaces in P4. The connections between this work and the work of Hassett, Hassett-Hyeon, Hyeon-Lee, and others, on the log minimal model program for the moduli space of curves will also be discussed. This is joint work with Radu Laza.