Summary: A Roman dominating function (RDF) on a graph \(G = (V,E)\) is defined to be a function \(f:V \rightarrow \{0,1,2\}\) 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 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 Restrained Roman dominating function on a graph \(G = (V,E)\) to be a function \(f:V \rightarrow \{0,1,2\}\) 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 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 \(\gamma_{\mathrm{rR}}(G)\). In this paper, we initiate a study of this parameter.