I will report on an ongoing joint project with David Helm and Yichao Tian. Let $p$ be a prime unramified in a totally real field $F$. The Goren-Oort strata are defined by the vanishing locus of the partial Hasse invariants; they furnish an analog of the stratification of modular curves mod $p$ by the ordinary locus and the supersingular locus. We give an explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for $F$. An interesting application of this result is that, when $p$ is inert of even degree, certain generalizations of the strata considered by Goren-Oort contribute non-trivially as Tate cycles to the cohomology of the special fiber of the Hilbert modular varieties. Under some mild conditions, they generate all the Tate cycles.