In this talk, we begin with the famous example due to Frey and Hellegouarch which attaches an elliptic curve to the Fermat equation. We will then explore other methods of attaching elliptic curves to Diophantine equations, for example, as demonstrated in the work of Dahmen. Finally, we will see a final example of such an attachment over a number field. This final example forms part of ongoing work in collaboration with Mike Bennett (UBC) and Samir Siksek(Warwick).