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Theta functions play a major role in many current researches and are powerfultools for studying integrable systems. The purpose of this paper is to providea short and quick exposition of some aspects of meromorphic theta functions forcompact Riemann surfaces. The study of theta functions will be done via ananalytical approach using meromorphic functions in the framework of Mumford.Some interesting examples will be given : the classical Kirchhoff equations inthe cases of Clebsch and Lyapunov-Steklov, the Landau-Lifshitz equation and thesine-Gordon equation.

Fonctions m\'eromorphes et fonctions th\^eta sur les surfaces de Riemann