Appears in collection : A conference in honor of Arthur Ogus on the occasion of his 70th birthday

I will describe the structure of finite morphisms between smooth Berkovich curves. The tame case is well known so the accent will be on the wild case. In particular, I will describe the loci of points of multiplicity n and their relation to Herbrand function and the ramification theory. If time permits I will also talk about the different function associated to a morphism. Slides : http://www.ihes.fr/~abbes/Ogus/temkin.pdf