The number of primes of a number field for which a smooth projective curve has bad reduction is known to be finite. Given a prescribed set of prime integers, S, one may ask how many smooth projective curves with fixed genus have bad reduction exactly at the primes in S. In this talk, we consider the question for genus 1 and genus 2 curves when S contains at most 2 primes. We survey some old and new results on how to list all isomorphism classes of such curves over the rational field.