Cubic threefolds were the first class of varieties who were shown to be unirational but not rational (Clemens,Griffiths). The key tool of the proof is the intermediate Jacobian, a principally polarized abelian variety of dimension 5. There is a second link to Hodge theory, namely via cubic fourfolds (Allcock, Carlson, Toledo) which leads to a $10$-dimensional ball quotient model. Looking at cubic threefolds from these different points of view leads to various geometrically relevant compactifications of the moduli space of cubic threefolds. In this talk I will discuss the geometry and the topology of these spaces. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.