  We compute and compare the (intersection) cohomology of various natural
geometric compactifications of the moduli space of cubic threefolds: the GIT
compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal
compactifications of the ball quotient model, due to Allcock-Carlson-Toledo.
Our starting point is Kirwan's method. We then follow by investigating the
behavior of the cohomology under the birational maps relating the various
models, using the decomposition theorem in different ways, and via a detailed
study of the boundary of the ball quotient model. As an easy illustration of
our methods, the simpler case of the moduli of cubic surfaces is discussed in
an appendix.
