The moduli space of smooth complex cubic surfaces can be compactified from the point of view of geometric invariant theory (GIT), and from the point of view of the ball quotient. The Kirwan desingularization resolves the GIT singularities to yield a smooth Kirwan compactification, while the toroidal compactification of the ball quotient is also smooth. We show that these two smooth compactifications are, however, not isomorphic, and study their birational geometry, and related classical algebro-geometric constructions in further detail. Based on joint works with S. Casalaina-Martin, K. Hulek, and R. Laza