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The work that consists of two parts is devoted to the problem of enumeratingunrooted $r$-regular maps on the torus up to all its symmetries. We begin withenumerating near-$r$-regular rooted maps on the torus, projective plane and theKlein bottle. We also present the results of enumerating some special kinds ofmaps on the sphere: near-$r$-regular maps, maps with multiple leaves and mapswith multiple root semi-edges. For $r=3$ and $r=4$ we obtain exact analyticalformulas. For larger $r$ we derive recurrence relations. Then using theseresults we enumerate $r$-regular maps on the torus up to homeomorphisms thatpreserve its orientation --- so-called sensed maps. Using the concept of aquotient map on an orbifold we reduce this problem to enumeration of certainclasses of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressionsfor the numbers of $r$-regular sensed maps by edges. All these results will beused in the second part of the work to enumerate $r$-regular maps on the torusup to all homeomorphisms --- so-called unsensed maps.

Enumeration of $r$-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps