Abstarct: In this thesis we study fuzzy graphs and some of it’s applications, and extend some properties of crisp graphs to fuzzy graphs. In the first chapter we state some basic definitions and properties of fuzzy graphs. In chapter 2, we study some operations on fuzzy graphs such as union, join, cartesian product, composition, normal product and tensfor product. In chapter 3 we study isomorphism on fuzzy graphs, and in chapter 4 we study regular and totally regular fuzzy graphs. In chapter 5 we consider planar fuzzy graphs and investigate Euler’s formula for fuzzy graphs. Finally in chapter 6, we state some applications of fuzzy graph such as eigenvalues and energy in fuzzy graph, coloring fuzzy graphs and dominition in fuzzy graph.