One of the important parameters within graph theory, in terms of applications and research attractiveness, is domination number. A subset D of vertices in graph G= ( V,E ), is called a dominating set of G if every vertex in V-D is adjacent to a vertex in D . The concept of domination in graphs, with its many variations, are now well studied in graph theory and there are many articles and books in this regard. One of the variations of this parameter is total restrained domination concept in graphs which is introduced in 2005 by De-Xinang Ma et al.. A subset D of vertices in graph G= ( V,E ), is called total restrained dominating set of G if D is a dominating set of G and the induced subgraphs D and V-D have no isolated vertex. The total restrained domination number of G , denoted by ? tr ( G ), is the minimum cardinality of a total restrained dominating set of G . In this thesis, among the study of the known results related to this concept, we defined some new concepts, which help us to provide some new bounds for the total restrained domination number of graphs.