Quadratic Points on $X_0(N)$

Mazur's celebrated theorem, together with follow-up work by Kenku and Momose, gives a complete classification of the rational points on the modular curves $X_0(N)$. This result is special to $\mathbb{Q}$, and no direct analogue holds over general number fields. In this talk, we restrict to the case of quadratic number fields $K$ and survey the current state of the art for understanding $X_0(N)(K)$. We describe the role of Atkin-Lehner involutions and Chabauty-type methods that together allow one to determine quadratic points on $X_0(N)$ in some cases.