Summary: A double Roman dominating function on a graph \(G\) with vertex set \(V(G)\) is defined by \textit{R. A. Beeler} et al. [Discrete Appl. Math. 211, 23--29 (2016; Zbl 1348.05146)] as a function \(f: V(G)\to\{0, 1,2,3\}\) having the property that if \(f(v)=0\), then the vertex \(v\) must have at least two neighbors assigned 2 under \(f\) or one neighbor \(w\) with \(f(w)= 3\), and if \(f(v)= 1\), then the vertex \(v\) must have at least one neighbor \(u\) with \(f(u)\geq 2\). A set \(\{f_1,f_2,\dots,f_d\}\) of distinct double Roman dominating functions on \(G\) with the property that \(\sum^d_{i=1} f_i(v)\leq 3\) for each \(v\in V(G)\) is called a double Roman dominating family (of functions) on \(G\). The maximum number of functions in a double Roman dominating family on \(G\) is the double Roman domatic number of \(G\), denoted by \(d_{dR}(G)\). We initiate the study of the double Roman domatic number, and we present different sharp bounds on \(d_{dR}(G)\). In addition, we determine the double Roman domatic number of some classes of graphs.