  Let $D$ be a strongly connected digraph. The average distance
$\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the
distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and
proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average
distances of the vertices of $D$, respectively. We obtain sharp upper and lower
bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and
describe the extreme digraphs for all the bounds. We also obtain such bounds
for strong tournaments. We show that for a strong tournament $T$, we have
$\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may
conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We
present an infinite family of non-regular strong digraphs $D$ such that
$\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well.
