Let \(E\) be an elliptic curve defined over the field of rational numbers and suppose that \(E\) does not have (potential) complex multiplication. For every prime \(p\), the action of the absolute Galois group of \(\mathbb{Q}\) on the \(p\)-torsion points of \(E\) gives rise to a representation \(\rho_{E,p}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{Aut}(E[p])\cong\mathrm{GL}_2(\mathbb{F}_p)\). Serre's celebrated open image theorem shows that this representation is surjective for all \(p\) greater than some bound \(p_0=p_0(E)\), depending in principle on \(E\). Serre has asked whether there is a universal \(p_0\) that works for all (non-CM) elliptic curves over \(\mathbb{Q}\). Building on results by many authors, including most recently Le Fourn and Lemos, we prove that for \(p>37\) there are at most two possibilities (up to conjugacy) for the image of \(\rho_{E, p}\): the whole group \(\mathrm{GL}_2(\mathbb{F}_p)\) and the normaliser of a so-called non-split Cartan subgroup. This is joint work with Lorenzo Furio.