Abstarct: For k ≥ 1, a k-fair dominating set in a graph G is a dominating set S such that |N(v)∩S| = k for every vertex v ∈ V \ S. The k-fair domination number of G, denoted by fdk(G), is the minimum cardinality of a kFD-set. A fair dominating set in G is a kFD-set for some integer k ≥ 1. The fair domination number of G, denoted by fd(G), that is not the empty graph is the minimum cardinality of an FD-set in G. An FD-set of G of cardinality fd(G) is called a fd(G)-set. In this thesis, we study the fair domination number. We present new upper bounds for the 1-fair domination number of cactus graph, unicyclic graph and outerplanar graph.We also characterize all graphs achieving equality for the upper bound.