Abstarct: In this dissertation, we study Cayley graphs on dihedral groups. First we present primary definitions and concepts of graph theory and group theory, and then will define Cayley graph, and classify the 4-valent one-regular normal Cayley graph G on a dihedral group whose vertex stabilizers in Aut (G) are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.