In this series of talks, I survey recent progress on the (stable) rationality problem for
smooth projective hypersurfaces and quadric bundles. I explain in some detail Voisins degeneration method and its modifications due to Colliot-Thélène–Pirutka and myself. In order
to apply this method, I recall some basic facts about unramified cohomology and explain
the known strategies to construct unirational examples with nontrivial unramified cohomology. The latter originated in the work of Artin–Mumford (1972) and Colliot-Thélène–Ojanguren (1989) and has more recently been used in high degree by Asok (2013) and myself.
Special emphasize will be given to an example of a quadric surface bundle over $\mathbb P^2$ with nontrivial unramified degree two cohomology due to Hassett–Pirutka–Tschinkel (2016), and its
generalizations to higher dimensions and higher degree unramified cohomology, found by