  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.
