Let E be an elliptic curve over an algebraically closed field k of
characteristic p>0. Then the physical p-torsion E[p](k) is either trivial,
and E is called supersingular, or E[p](k) is a group of order p. More
generally, if X/k is an abelian variety of dimension g, then X[p](k)
is isomorphic to (Z/p)^f for some number f, called the p-rank of X.
The p-rank induces a stratification of the moduli space of abelian
varieties; via the Torelli functor, it induces a stratification of the
moduli space of (hyperelliptic) curves.
I'll discuss recent results on the geometry of these strata, with
special attention to their structure at the boundary of the moduli
space. This information yields new applications about the prime-to-p
part of the class group of a quadratic function field with specified geometric
p-rank; the existence of absolutely simple hyperelliptic Jacobians of
every p-rank; and the stratification of the moduli space of curves by
Newton polygon.