Abstract
This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois action on the Picard group, Brauer group, higher unramified cohomology, global differentials, specialisation method (via R-equivalence), geometrically rational surfaces, cubic hypersurfaces.
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