Let $G$ be an open subgroup of $GL_2(\hat{\mathbb{Z}})$ satisfying $\operatorname{det}(G)=\hat{\mathbb{Z}}^{\times}$ and $-I\in G$. Associated to $G$, there is a modular curve $X_G$ defined over $\mathbb{Q}$ which parametrizes elliptic curves with $G$-level structure. In Part I, we give a classification of modular curves with genus equal to a fixed nonnegative integer $g$. In particular, we show that all modular curves of genus $g$ lie in finitely many families of $\mathbb{Q}^{\operatorname{ab}}$-twists of modular curves. We also describe an algorithm for computing all families of modular curves of a fixed genus $g$ and use this to compute projective models for these modular curves. This algorithm has been fully implemented for $g \leq 12$. In Part II, we identify all families of geometrically hyperelliptic modular curves. By studying the existence of rational points on certain twists of a genus 0 curves, we characterize hyperelliptic modular curves within geometrically hyperelliptic families. We also classify modular curves of genus $0$ up to $\mathbb{Q}$-isomorphism.