Cubic threefolds have played an important role in algebraic geometry ever since Clemens and Griffiths showed that these are unirational but not rational varieties. The crucial ingredient in their proof is a careful analysis of the intermediate Jacobian. The map which associates to a cubic threefold $X$ its intermediate Jacobian $IJ(X)$ defines an injective map from the GIT moduli space of smooth cubics to the moduli space $A_5$ of principally polarized abelian $5$-folds. Here we shall discuss the behaviour of this map when the cubic threefold acquires singularities. Our main result is that the intermediate Jacobian map extends to a regular morphism $\overline{IJ}: \widetilde M \to A_5^{\operatorname{Vor}}$ from the wonderful blow-up of the GIT moduli space of cubic threefolds to the second Voronoi compactification. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.