In this joint work with Ben Moonen a generalization of the Ekedahl-Oort stratification is described: For every abelian scheme over a base scheme $S$ of characteristic $p$, Ekedahl and Oort defined a stratification of $S$. This has been in particular succesfully applied to universal abelian schemes over moduli spaces of abelian varieties. We generalize this to arbitrary proper and smooth schemes of Hodge type. It turns out that we get stratifications which are indexed by certain quotients of a symmetric group. There is also the notion of this stratification with $G$-structure for $G$ a reductive group and using results of Lusztig we get a stratification indexed by a certain quotient of the Weyl group of $G$.