Shimura curves are generalizations of modular curves, where the matrix ring is replaced by a quaternion algebra over a totally real field. Recently, Long, Maclachlan, and Reid proved that the number of Shimura curves of bounded genus is finite. In this talk, we describe a method to explicitly enumerate all Shimura curves of genus at most 2; along the way, we tabulate all totally real number fields of root discriminant at most 14. We examine some of the mathematically and computationally interesting aspects of each of these tasks in turn.