Circulants are Cayley graphs for cyclic groups, and admit dihedral symmetries. This talk is a summary of some work done with a research student (from Spain) on the genus spectrum of embeddings of circulants on orientable surfaces. First, we derived a formula for the {\em maximum genus\/} of such embeddings. Then we turned to the question of finding the {\em minimum\/} genus of such embeddings, for various classes of circulants. In doing this, we found several counter-examples to a claimed theorem by Costa, Strapasson, Alves and Carlos (2010) on embeddings of circulants on the torus (genus 1), and then we went on to determine all circulants that have minimum genus 1 or 2. This work involved a combination of mathematics and computer experimentation, some of which will be described, with illustrations.