The first step of the course is to see the following: any birational map of the plane can be decomposed into the blow-up of a point\, elementary transformations between Hirzebruch surfaces and the exchange of the two fibrations of P1�P1. So\, the Fano surface P2 and the Hirzebruch surfaces\, and elementary birational maps between them\, are the base of birational maps of the plane. The second step of the course is to generalise this concept to non-closed fields. The base-stones will be minimal del Pezzo surface and minimal conic fibrations over the projective line\, and elementary birational maps between them.