Cubic threefolds have been studied from various points of view. Clemens and Griffiths proved in 1972 that a smooth cubic threefold is unirational, but not rational, thus providing the first example of this phenomenon. The moduli spaces of cubic threefolds (with mild singularities) were studied by many authors, notably Allcock, Carlson Toledo and many others. Here I will report on the geometry and topology of the GIT moduli space of cubic threefolds and its smooth models such as the Kiran blow-up of the toroidal compactification of the ball quotient model. This is joint work with Casalaina-Martin, Grushevsky and Laza.