Compositions and relations in the Cremona groups

The Cremona group is the group of biraitonal symmetries of the projective space of dimension n. After an introduction, the thesis contains the proof that the complex Cremona group of the plane is compactly presented. It also presents the abelianisation of the real Cremona group of the plane. The last part consists of a speculative look at specific Cremona transformations of higher dimensional projective spaces, the so called punctual transformations.