Abstract: A subset $$S\subseteq V$$ is a dominating set in a graph $$G = (V,E)$$ if every vertex $$v \in V \setminus S$$ is adjacent to at least one vertex $$v\in S$$. The minimum cardinality of a dominating set is called the domination number of G and it is denoted by $$\gamma(G)$$. In this thesis, we study the domination number for standard graphs and derive some upper and lower bounds for $$\gamma(G)$$. The main object of this thesis is to study dominating sets.