Let be an elliptic curve over , without complex multiplication over . For a prime number, consider the representation induced by the Galois action on the group of -torsion points of . A theorem of Serre, published in 1972, asserts that there exists an integer such that the above representation is surjective for larger than . Serre then asked the following question: can be chosen independently of ? The classification of maximal subgroups of shows that this boils down to proving the triviality, for large enough , of the sets of rational points of four families of modular curves, namely , , and (we say that a point of one of those curves is trivial if it is either a cusp, or the underlying isomorphism class of elliptic curves has complex multiplication over ). The (so-called exceptional) case of was ruled out by Serre. The fact that is made of only cusps for is a well-known theorem of Mazur. In this talk we will present a proof that is trivial for large enough (joint work with Yuri Bilu).