Meromorphic Functions and Theta Functions on Riemann Surfaces

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