  Let $F_1, F_2, ..., F_k$ be graphs with the same vertex set $V$. A subset $S
\subseteq V$ is a simultaneous dominating set if for every $i$, $1 \le i \le
k$, every vertex of $F_i$ not in $S$ is adjacent to a vertex in $S$ in $F_i$;
that is, the set $S$ is simultaneously a dominating set in each graph $F_i$.
The cardinality of a smallest such set is the simultaneous domination number.
We present general upper bounds on the simultaneous domination number. We
investigate bounds in special cases, including the cases when the factors,
$F_i$, are $r$-regular or the disjoint union of copies of $K_r$. Further we
study the case when each factor is a cycle.
