Lectures on the Mordell-Weil Theorem

This is a translation of "Auto ur du theoreme de Mordell-Weil", a course given by J . -P. Serre at the College de France in 1980 and 1981. These notes were originally written weekly by Michel Waldschmidt and have been reproduced by Publications Mathematiques de l'Universite de Paris VI, by photocopying the handwritten manuscript. The present translation follows roughly the French text, with many modi- fications and rearrangements.