  We study the birational geometry of the moduli space of complete $n$-quadrics
$X$. We exhibit generators for Eff$(X)$ and Nef$(X)$, the cone of effective
divisors and the cone of nef divisors, respectively. As a corollary X is a Mori
Dream space. Furthermore, we run the MMP on $X$ and find a moduli
interpretation for the models induced by the generators of the nef cone. In the
case of complete quadric surfaces, we describe all the birational models of $X$
induced by the movable cone and find a moduli interpretation for some of these
models.
