Disproving stable rationality

In the last four years, a series of papers by several authors has established that some very classical, rationally connected, complex varieties (cyclic covers of projective space with ramification locus of low degree, hypersurfaces of low degree, quadric bundles over rational varieties) are not stably birational to projective space. In the first part of the talk I shall give a general description of the method and of some of its variants, and I shall try to list the main results achieved. The second part of the talk will be devoted to a recent variant of the technique.