Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Artur Travesa i Grau[en] We start this thesis with a brief study on the Rieman-Roch Theorem so we can later introduce the concept of elliptic curve. We’ll proceed studying these as Weirstrass plane cubics and their reduction behaviour. Subsequently we’ll develop the construction of Frey’s curve and study some of its properties. Then, we give a short introduction to modular functions and Galois representation. Finally, we draw an outline for the proof of Fermat’s Theorem, where we can appreciate the importance of said curve. We conclude with an application of this method on other diofantic equations.