Summary: A subset \(D\) of vertices of a graph \(G\) is a global total \(k\)-dominating set if \(D\) is a total \(k\)-dominating set of both \(G\) and \(\overline{G}\). The global total \(k\)-domination number of \(G\) is the minimum cardinality of a global total \(k\)-dominating set of \(G\) and it is denoted by \(\gamma_{k t}^g(G)\). In this paper we introduce this concept and we begin the study of its mathematical properties. Specifically, we prove that the complexity of the decision problem associated to the computation of the value \(\gamma_{k t}^g(G)\) is NP-complete. Moreover, we present tight bounds for this parameter and we obtain relationships between the global total \(k\)-domination number of \(G\) and the total \(k\)-domination numbers of \(G\) and \(\overline{G}\), and other parameters of the graph \(G\).