  We recall some properties, unfortunately not all, of the Cremona group.
  We first begin by presenting a nice proof of the amalgamated product
structure of the well-known subgroup of the Cremona group made up of the
polynomial automorphisms of $\mathbb{C}^2$. Then we deal with the
classification of birational maps and some applications (Tits alternative,
non-simplicity...) Since any birational map can be written as a composition of
quadratic birational maps up to an automorphism of the complex projective
plane, we spend time on these special maps. Some questions of group theory are
evoked: the classification of the finite subgroups of the Cremona group and
related problems, the description of the automorphisms of the Cremona group and
the representations of some lattices in the Cremona group. The description of
the centralizers of discrete dynamical systems is an important problem in real
and complex dynamic, we make a state of art of this problem in the Cremona
group.
  Let $Z$ be a compact complex surface which carries an automorphism $f$ of
positive topological entropy. Either the Kodaira dimension of $Z$ is zero and
$f$ is conjugate to an automorphism on the unique minimal model of $Z$ which is
either a torus, or a K3 surface, or an Enriques surface, or $Z$ is a
non-minimal rational surface and $f$ is conjugate to a birational map of the
complex projective plane. We deal with results obtained in this last case:
construction of such automorphisms, dynamical properties (rotation domains...)
are touched on.
