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This paper presents our preliminary results on regular meshes in which all faces have the same size and all vertices have the same valence. A regular mesh is denoted by (n, m, g) where n is the number of the sides of faces, m is the valence of vertices and g is the genus of the mesh. For g = 0, regular meshes include regular platonic solids, all two sided polygons. For g = 1 regular meshes include regular tilings of infinite plane. Our work shows that there exist infinitely many regular meshes for g 1. Moreover, we have constructive proofs that describe how to create high genus regular meshes that consist of triangles and quadrilaterals (3, m, g) and (4, m, g).