  Moduli spaces of complete collineations are wonderful compactifications of
spaces of linear maps of maximal rank between two fixed vector spaces. We
investigate the birational geometry of moduli spaces of complete collineations
and quadrics from the point of view of Mori theory. We compute their effective,
nef and movable cones, the generators of their Cox rings, and their groups of
pseudo-automorphisms. Furthermore, we give a complete description of both the
Mori chamber and stable base locus decompositions of the effective cone of the
space of complete collineations of the 3-dimensional projective space.
