Abstarct: A set S⊆V of vertices in a graph G = (V,E) is a dominating set if every vertex in V – S is adjacent to a vertex in S. The minimum cardinality of a dominating set is called the domination number of G and is denoted by γ(G) . A dominating set which intersects every maximum independent set in G is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of G and is denoted by γ_it (G). In this thesis, we investigate this parameter. We determine the exact values of the independent transversal domination number in several families of graphs including paths, cycles and wheels. We also obtain different bounds for this parameter and study the complexity of it. Also, we answer some problems on this parameter. We obtain several new bounds for the independent transversal domination number of a graph, and characterize all graphs achieving equality for new bounds.