The space of complete collineations is a compactification of the space of matrices of fixed dimension and rank, whose boundary is a divisor with normal crossings. It was introduced in the 19th century and has been used to solve many enumerative problems. We show that this venerable space can be understood using the latest quotient constructions in algebraic geometry. Indeed, there is a detailed analogy between the complete collineations and the moduli space of stable pointed curves of genus zero. The remarkable results of Kapranov exhibiting the latter space as a Chow quotient, Hilbert quotient, and so on, all have counterparts for complete collineations. This analogy encompasses Vainsencher's construction of the complete collineations, as well as a form of the Gel'fand-MacPherson correspondence. There is also a tangential relation with the Gromov-Witten invariants of Grassmannians. The symmetric and anti-symmetric versions of the problem are considered as well. An appendix explains the original motivation, which came from the space of broken Morse flows for the moment map of a circle action.