  We survey basic properties and bounds for $q$-equivelar and $d$-covered
triangulations of closed surfaces. Included in the survey is a list of the
known sources for $q$-equivelar and $d$-covered triangulations. We identify all
orientable and non-orientable surfaces $M$ of Euler characteristic
$0>\chi(M)\geq -230$ which admit non-neighborly $q$-equivelar triangulations
with equality in the upper bound
$q\leq\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. These
examples give rise to $d$-covered triangulations with equality in the upper
bound $d\leq2\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. A
generalization of Ringel's cyclic $7{\rm mod}12$ series of neighborly
orientable triangulations to a two-parameter family of cyclic orientable
triangulations $R_{k,n}$, $k\geq 0$, $n\geq 7+12k$, is the main result of this
paper. In particular, the two infinite subseries $R_{k,7+12k+1}$ and
$R_{k,7+12k+2}$, $k\geq 1$, provide non-neighborly examples with equality for
the upper bound for $q$ as well as derived examples with equality for the upper
bound for $d$.
