The study of Galois representations attached to elliptic curves is a very fruitful branch of number theory, leading to the solution of very difficult problems, such as Fermat’s Last Theorem. Given an elliptic curve \(E\) defined over a number field \(K\), the representation \(\rho_{E,p}\) is described by the action of the absolute Galois group of \(K\) on the \(p\)-torsion points of \(E\). In 1972 Serre proved that for every elliptic curve \(E\) without complex multiplication there exists an integer \(N_E\) such that, for every prime \(p>N_E\), the Galois representation \(\rho_{E,p}\) is surjective onto \(\mathrm{GL}_2(\mathbb{F}_p)\). In the same article, he asked whether the constant \(N_E\) can be taken to be independent of the curve, and this became known as Serre’s Uniformity Question. In this talk, I will discuss the current progress towards an answer to this question in the case \(K=\mathbb{Q}\), and I will present the more general problem of classifying all the possible images of the adelic representation of \(E\), known as Mazur's program B.