  Acyclic and cyclic orientations of an undirected graph have been widely
studied for their importance: an orientation is acyclic if it assigns a
direction to each edge so as to obtain a directed acyclic graph (DAG) with the
same vertex set; it is cyclic otherwise. As far as we know, only the
enumeration of acyclic orientations has been addressed in the literature. In
this paper, we pose the problem of efficiently enumerating all the
\emph{cyclic} orientations of an undirected connected graph with $n$ vertices
and $m$ edges, observing that it cannot be solved using algorithmic techniques
previously employed for enumerating acyclic orientations.We show that the
problem is of independent interest from both combinatorial and algorithmic
points of view, and that each cyclic orientation can be listed with
$\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost
of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of
$\tilde{O}(m)$ time.
