  This article gives the proof of results announced in [J. Blanc, Finite
Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris,
S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational
surfaces.
  Given a finite Abelian subgroup of the Cremona group of the plane, we provide
a way to decide whether it is birationally conjugate to a group of
automorphisms of a minimal surface.
  In particular, we prove that a finite cyclic group of birational
transformations of the plane is linearisable if and only if none of its
non-trivial elements fix a curve of positive genus. For finite Abelian groups,
there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z,
whose non-trivial elements do not fix a curve of positive genus but which is
not conjugate to a group of automorphisms of a minimal rational surface.
  We also give some descriptions of automorphisms (not necessarily of finite
order) of del Pezzo surfaces and conic bundles.
