Let G = (V, μ, σ) be a fuzzy graph on a finite set V. A fuzzy subset μ′ of μ is called a fuzzy dominating set of G if, μ ′ ( v ) + ∑ x ∈ V ( σ ( x , v ) ∧ μ ′ ( x ) ) ≥ μ ( v ) for every v ∈ V . Fuzzy domination number γfz is defined accordingly. In this paper we initiate a study of this parameter. Some properties of fuzzy dominating sets are studied and fuzzy domination number γfz is determined for some graphs.