Let G = (V, E) be a graph. Let S be the set of all minimal total dominating sets of G. The total dominating graph D t (G) of G is the graph with the vertex set V∪S in which two vertices u and v are adjacent if u ∈ V and v is a minimal total dominating set of G containing u. In this paper, some properties of this new graph are obtained. Also characterizations are given for graphs (i) whose total dominating graphs are complete bipartite, (ii) whose total dominating graphs are Eulerian.