Keerthi Madapusi Pera - Boston College We will review the different aspects of the Siegel modular variety: its incarnation as a locally symmetric complex manifold; its moduli interpretation as a parameter space for abelian varieties, which allows one (where 'one' = 'Mumford') to construct a model for it over the integers; and new phenomena that arise in finite characteristic. In particular, we will see that there is a canonical Zariski dense open subspace, the ordinary locus, in characteristic p, defined essentially as the locus where the family of abelian varieties has the maximal number of p-torsion points.