As is classically known, one can put a group law on the set of points of an elliptic curve. When working in characteristic p, elliptic curves then break into two classes -- ordinary and supersingular -- depending on the p-torsion of these groups. In this talk, I will generalize this idea to hyperelliptic curves and look at the different classes that occur in higher genera. In particular, I will show that all possible p-ranks occur for each genus, and that they occur with the expected codimension when p>2.