Mazur's isogeny theorem states that if $p$ is a prime for which there exists an elliptic curve $E / \mathbb{Q}$ that admits a rational isogeny of degree $p$, then $p \in \{2,3,5,7,11,13,17,19,37,43,67,163 \}$. This result is one of the cornerstones of the theory of elliptic curves and plays a crucial role in the proof of Fermat's Last Theorem. In this expository paper, we overview Mazur's proof of this theorem, in which modular curves and Galois representations feature prominently.

Minor changes addressing referee suggestions. To appear in Proceedings of The Year-Long Program on Triangle Groups, Belyi Uniformization, and Modularity: IInd Trimester Proceedings, Bhaskaracharya Pratishthana, Pune, India (https://sites.google.com/view/bms2021/proceedings)