A textit{Roman dominating function} (RDF) on a graph $G = (V,E)$ is defined to be a function $ f:V rightarrow lbrace 0,1,2rbrace$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A set $S subseteq V$ is a textit{Restrained dominating set} if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. We define a textit{Restrained Roman dominating function} on a graph $G = (V,E)$ to be a function $f : V rightarrow lbrace 0,1,2 rbrace$ satisfying the condition that every vertex $u$ for which $f(u) = 0 $ is adjacent to at least one vertex $v$ for which $f(v)=2$ and at least one vertex $w$ for which $f(w) = 0$. The textit{weight} of a Restrained Roman dominating function is the value $f(V)= sum _{u in V} f(u)$. The minimum weight of a Restrained Roman dominating function on a graph $G$ is called the Restrained Roman domination number of $G$ and denoted by $gamma_{rR}(G)$. In this paper, we initiate a study of this parameter.