Programme:

1. Rationality, stable rationality, retract-rationality, unirationality, rational connectedness.
2. Examples of rational varieties. Rationality of intersection of two quadrics.
3. Birational invariants. Resolution of singularities and weak decomposition theorem. Holomorphic contravariant tensors.
4. Castelnuovo's rationality criterium.
5. Rationality of surfaces over non-algebraically-closed fields. Del Pezzo fibrations.
6. Conic bundles, discriminant. Double covers and Prym varieties. Intermediate Jacobian of a conic bundle.
7. Artin-Mumford's example of stably irrational unirational threefold. Torsion in homology. Brauer group.
8. Clemens-Griffiths's proof of irrationality of a smooth cubic threefold. Variety of lines on a cubic hypersurface. Weil's intermediate Jacobian and Griffiths's component.
9. Iskovskikh-Manin's proof of irrationality of a smooth quartic threefold. Method of maximal singularities. Birational rigidity.
10. Kollár's method: holomorphic forms in finite characteristic.
11. Voisin's degeneration method. Stable irrationality of a very general double cover of three-space branched in a quartic.
12. Beauville's proofs using Voisin's degeneration method.
13. Work of Colliot-Thélène and Pirutka. Stable irrationality of a very general quartic threefold.
14. Stable irrationality of a very general quartic fourfold after Totaro.