Abstarct: One of the important issues in graph theory is the study of the Ramsey numbers of graphs. For two graphs G and H, the Ramsey number R(G,H) is the smallest integer n, such that for any graph F of order n, either graph F contains G or its complement contains H. In this thesis we study different types of Ramsey numbers on graphs. We present results and bounds for the Ramsey number of pairs of graphs including paths, cycles, wheels and several other families of graphs.