Let $G$ be an open subgroup of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$. There is a modular curve $X_G$ which loosely parametrizes elliptic curves whose image of the Galois representation lies in $G$. In this talk, we give a classification of modular curves of low genus; this is achieved by studying twists of modular curves. We also describe an algorithm for computing all families of modular curves of a fixed genus and for computing explicit models for them.