  In this paper a number of results on cycles on the moduli space of
principally polarized abelian varieties is presented. Results include a
determination of the tautological ring, bounds on the order of torsion of the
top Chern class $\lambda_g$ and a determination of the cycle classes of the
Ekedahl-Oort stratification in characteristic $p$. It includes as special cases
formulas for the classes of loci like the $p$-rank $\leq f$ locus or the
$a$-number $\geq a$-locus. The results on the tautological ring are my own
work, the results on the torsion of $\lambda_g$ and on the cycle classes of the
Ekedahl-Oort stratification are joint work with Torsten Ekedahl and some of the
results on curves are joint work with Carel Faber.
