Picard curves are genus 3 curves of the form y^3=f(x), where f(x) is a polynomial of degree 4. They are the simplest non-hyperelliptic curves. This talk will discuss recent work with Chris Rasmussen, in which we found all Picard curves defined over the rationals with good reduction at all primes except p=3. This work was inspired by Nigel Smart´s enumeration of genus 2 curves with good reduction at all primes except p=2. This work is relevant to the study of modular curves, and employs several powerful tools, including Baker´s method and the LLL algorithm.