In joint work with K. Kedlaya and J. Lau, we compile a complete list of isomorphism class representatives of curves of genus 6 over the finite field of two elements. We use explicit descriptions of canonical curves in each stratum of the Brill-Noether stratification of the moduli space of genus 6 curves, due to Mukai in the generic case. In this talk, we will discuss some motivations and applications for having a census of low genus curves over various finite fields, including in the study of the moduli space of curves. We will also mention how recent work of Bergstrom-Canning-Petersen-Schmitt helps with independently verifying the completeness of our census.