Automorphisms of rational surfaces

A rational surface $X$ is a projective surface which is birationally equivalent to the projective plane (examples are obtained by blowing up a finite number of points of the projective plane). The group of all regular and invertible transformations $f\colon X \to X$ is the group of automorphisms of $X$. There are interesting questions regarding this group: For which surfaces is it infinite? How big can it be? What is the typical dynamical behaviour of automorphisms?$\dots$ I shall describe some of the main examples, together with a few open questions.