  With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class
of vertex-transitive triangulations of surfaces. In particular, there are
infinite series of cyclic $q$-equivelar triangulations of orientable and
non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq
3$. Series of cyclic tessellations of surfaces are derived from these
triangulated series.
