Abstarct: In this thesis, we study the transversal number of graphs, and state some features, applications and bounds available for this parameter. In order to emphasize the practical importance of this problem, we present some efficiently computable bounds in this thesis. Then, we will improve some of these bounds and present new bounds for graphs and connected graphs that have a non-cut-vertex of maximum degree. Also, we characterize the new bounds. Finally, we present a constructive characterization of bipartite graphs with specified transversal number.