  For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at
least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman
$k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there
exist at least $k$ and for any vertex $v$ with $f(v)\neq 0$ at least $k-1$
vertices in its neighborhood with $f(w)=2$. The minimum weight of a Roman
$k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple
domination number of the graph where the weight of $f$ is $f(V)=\sum_{v\in
V}f(v)$.
  In this paper, we initiate to study the Roman $k$-tuple domination number of
a graph, by giving some sharp bounds for the Roman $k$-tuple domination number
of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding
the Roman $k$-tuple domination number of some known graphs is our other goal.
Some of our results extend these one given by Cockayne and et al.} in 2004 for
the Roman domination number.
