Consider a non-CM elliptic curve $E/\mathbb{Q}$. The natural Galois action on the torsion points of $E$ can be encoded by a Galois representation $$\rho_E : \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}_2(\widehat{\mathbb{Z}}).$$ A famous theorem of Serre say that the image of $\rho_E$ is an open, and hence finite index, subgroup of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$. We shall describe recent results that allow us to compute the image of $\rho_E$ for any $E/\mathbb{Q}$.