Summary: Let \(G\) be a simple graph with vertex set \(V\). A double Roman dominating function (DRDF) on \(G\) is a function \(f:V\rightarrow\{0,1,2,3\}\) satisfying that if \(f(v)=0\), then the vertex \(v\) must be adjacent to at least two vertices assigned \(2\) or one vertex assigned \(3\) under \(f\), whereas if \(f(v)=1\), then the vertex \(v\) must be adjacent to at least one vertex assigned \(2\) or \(3\). The weight of a DRDF \(f\) is the sum \(sum_{v\in V}f(v)\). A total double Roman dominating function (TDRDF) on a graph \(G\) with no isolated vertex is a DRDF \(f\) on \(G\) with the additional property that the subgraph of \(G\) induced by the set \(\{v\in V:f(v)\neq 0\}\) has no isolated vertices. The total double Roman domination number \(\gamma_{\mathrm{tdR}}(G)\) is the minimum weight of a TDRDF on \(G\). In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.