We will report current progress on the reduction of Hilbert-Blumenthal varieties. We determine the dimension and singularities of the strata defined by various p-adit invariants. The invariants include Lie types, a-numbers and leaves. We will also explain the connection with the Hecke orbit problems and the irreducibility of the non-supersingular strata. We can prove the irreducibility of these non-supersingular strata in many cases, including the cases when p is unramified or totally ramified. For the supersingular strata, the number of irreducible components is determined and is expressed in terms of special values of the zeta function. This is a joint work with Ching-Li Chai.