With Yongyuan Huang and Jun Bo Lau, we recently completed a census of genus-6 curves over the field F_2, and are working on a similar census in genus 7. This uses Mukai's "flowcharts" for describing canonical curves in this genera. We discuss some of the key features of this classification; some aspects of computational group theory required to convert this classification into tractable computations; and some applications of the results, including relative class number problems for function fields, gonality of curves over finite fields (work of
Faber-Grantham-Howe), and cohomology of modular curves (work of Canning-Larson and Bergstrom-Canning-Petersen-Schmitt).