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We review the author's results on Mather's $\beta$ function : non-strictconvexity of $\beta$ when the configuration space has dimension two, linkbetween the size of the Aubry set and the differentiability of $\beta$,correlation between the rationality of the homology class and thedifferentiability of $\beta$, equality of the Mather set and the Aubry set fora large number of cohomology classes when the configuration space has dimensiontwo, link beween the differentiability of $\beta$ and the integrability of thesystem. Ma\~{n}\'e's conjectures are discussed in Chapters 6 and 7. A shortlist of open problems is given at the end of each chapter. In Appendix A weprove a theorem which extends Theorem 5 of reference [Mt09]. In Appendix B wediscuss a geometrical problem which arises from Chapter 3, but may be ofindependant interest.

Syst\`emes lagrangiens et fonction $\beta$ de Mather