  In the first part of this text we give a survey of the properties satisfied
by the C1-generic conservative diffeomorphisms of compact surfaces. The main
result that we will discuss is that a C1-generic conservative diffeomorphism of
a connected compact surface is transitive. It is obtain as a consequence of a
connecting lemma for pseudo-orbits. In the last parts we expose some recent
developments of the C1-perturbation technics and the proof of this connecting
lemma. We are not aimed to deal with technicalities nor to give the finest
available versions of these results. Besides this theory exists also in higher
dimension and in the non-conservative setting, we restricted the scope of this
presentation to the conservative case on surfaces, since it offers some
simplifications which allow to explain in an easier way the main ideas of the
subject.
