We study the p-torsion group scheme of Abelian varieties of dimension g over algebraically closed fields. In characteristic 0, the only possibility for the isomorphism class of the p-torsion group scheme is the product of 2g copies of (Z/pZ). In characteristic p, this is radically different. Since multiplication by p is inseparable, the kernel is smaller and there are 2^g distinct possibilities, called Ekedahl-Oort types, for the isomorphism class of the p-torsion group scheme. The smallest-dimensional example of this is the distinction between ordinary and supersingular elliptic curves. We are interested in which Ekedahl-Oort types occur if we put restrictions on the Abelian varieties we study. We survey known results and present new results, specifically if the Abelian variety is the Jacobian of an Artin-Schreier cover or if the Abelian variety has unitary structure with signature (m,2).

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