  For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity}
between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of
internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of
$G$, denoted $\overline{\kappa}(G),$ is the average of $\kappa_G(u,v)$ taken
over all unordered pairs of distinct vertices $u,v$ of $G$. Analogously, for a
directed graph $D$, the {\em connectivity} from $u$ to $v$, denoted
$\kappa_D(u,v)$, is the maximum number of internally disjoint directed $u$--$v$
paths in $D$. The {\em average connectivity} of $D$, denoted
$\overline{\kappa}(D)$, is the average of $\kappa_D(u,v)$ taken over all
ordered pairs of distinct vertices $u,v$ of $D$. An {\em orientation} of a
graph $G$ is a directed graph obtained by assigning a direction to every edge
of $G$. For a graph $G$, let $\overline{\kappa}_{\max}(G)$ denote the maximum
average connectivity among all orientations of $G$. In this paper we obtain
bounds for $\overline{\kappa}_{\max}(G)$ and for the ratio
$\overline{\kappa}_{\max}(G)/\overline{\kappa}(G)$ for all graphs $G$ of a
given order and in a given class of graphs. Whenever possible, we demonstrate
sharpness of these bounds. This problem had previously been studied for trees.
We focus on the classes of cubic $3$-connected graphs, minimally $2$-connected
graphs, $2$-trees, and maximal outerplanar graphs.
