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A {\it regular balanced Cayley map} (RBCM for short) on a finite group$\Gamma$ is an embedding of a Cayley graph on $\Gamma$ into a surface, withsome special symmetric property. People have classified RBCM's for cyclic,dihedral, generalized quaternion, dicyclic, and semi-dihedral groups. In thispaper we classify RBCM's on the group ${\rm PSL}(2,p)$ for each prime number$p>3$.

Regular balanced Cayley maps on ${\rm PSL}(2,p)$