Deciding whether a given algebraic variety is rational, or birational to projective space, is an age-old and challenging problem in algebraic geometry. The rationality problem for rationally connected varieties has seen incredible advances in the last several years,thanks to a degeneration method for the Chow group of 0-cycles initiatedby Voisin, developed by Colliot-Thélène and Pirutka, and recently refined by Schreieder. After summarizing some of these advances, I will speak about joint work with Christian Boehning and Alena Pirutka on the rationality problem for two types of Fano fourfolds lying on the boundary of where different techniques are required: hypersurfaces of bidegree (2,3) in P2 x P3 and complete intersection of type (2,3) in P6. The first haveindex 1 and Picard rank 2, and we prove that the very general such hypersurface is not stably rational by exploiting conic bundle and cubic surface bundle structures. The second have index 2 and Picard rank 1, and are more challenging.