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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, we show that   $X$   is a Fano variety. Furthermore, we run the Minimal Model Program 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.

Birational Geometry of the Space of Complete Quadrics