The structure of p-torsion, or of p-power-torsion on abelian varieties in characrteristic p leads to several stratifications and foliations of the moduli space. We give a survey of these structures; we discuss proofs of conjectures by Manin and by Grothendieck using these results. One of the essential ingredients is the "Purity Theroem" (joint work with A. J. de Jong).