Resolution of singularities by p-alterations

de Jong proved that any variety X can be desingularized by an alteration f:X’–>X, i.e. a proper surjective generically finite morphism. This was strengthened by Gabber as follows: f can be chosen of degree prime to a fixed prime l invertible on X.In this talk, I’ll tell about the most recent progress on the subject: if X is of finite type over a quasi-excellent threefold then one can desingularize X by an alteration whose degree is only divisible by primes non-invertible on X. We will also discuss finer results that deal with divisors and desingularize morphisms in the sense of semistable reduction.