Recent developments of the Luroth problem

A complex  irreducible algebraic  variety X  is unirational, resp. rational,  if there is a many-to-one resp. one-to-one,  surjective map from projective space to X.  For curves and for surfaces, the two notions coincide (Lueroth 1876, Castelnuovo 1893). In dimension at least 3, they differ (Clemens-Griffiths, Iskovskikh-Manin, Artin-Mumford, 1974; Koll'ar 1995). However, for many natural classes of unirational varieties, rationality remains  an open question. In 2013, Voisin used a specialization method to disprove rationality of some varieties. Her method also disproves stable rationality. It has been quickly developed, and applied to more classes of varieties.