Let G be an open subgroup of GL_2 of Z-hat. 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 (over rationals) 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. We will also talk about generalization to the number field case, and potential applications.