Hypersurfaces that are not stably rational

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes Colliot-Thelene and Pirutka's theorem that very general quartic 3-folds are not stably rational. The result covers all the degrees in which Kollar proved that a very general hypersurface is non-rational, and a bit more. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.