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This text is an exposition of non-Archimedean curves and Schottkyuniformization from the point of view of Berkovich geometry. It consists of twoparts, the first one of an introductory nature, and the second one moreadvanced. The first part is meant to be an introduction to the theory ofBerkovich spaces focused on the case of the affine line. We define theBerkovich affine line and present its main properties, with many details:classification of points, path-connectedness, metric structure, variation ofrational functions, etc. Contrary to many other introductory texts, we do notassume that the base field is algebraically closed. The second part is devotedto the theory of Mumford curves and Schottky uniformization. We start bybriefly reviewing the theory of Berkovich curves, then introduce Mumford curvesin a purely analytic way (without using formal geometry). We define Schottkygroups acting on the Berkovich projective line, highlighting how geometry andgroup theory come together to prove that the quotient by the action of aSchottky group is an analytic Mumford curve. Finally, we present an analyticproof of Schottky uniformization, showing that any analytic Mumford curves canbe described as a quotient of this kind. The guiding principle of ourexposition is to stress notions and fully prove results in the theory ofnon-Archimedean curves that, to our knowledge, are not fully treated in othertexts.

Berkovich curves and Schottky uniformization