  This papers focuses on the average order of dominating sets of a graph. We
find the extremal graphs for the maximum and minimum value over all graphs on
$n$ vertices, while for trees we prove that the star minimizes the average
order of dominating sets. We prove the average order of dominating sets in
graphs without isolated vertices is at most $3n/4$, but provide evidence that
the actual upper bound is $2n/3$. Finally, we show that the normalized average,
while dense in $[1/2,1]$, tends to $\frac{1}{2}$ for almost all graphs.
