Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and $x,y{in}X$ ($x{ eq}y$) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ${Delta}(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(${Delta}(G)$) is abelian if and only if ${mid}G{mid}{leq}2$; ${mid}Aut({Delta}(G)){mid}$ is of prime power if and only if ${mid}G{mid}{leq}2$, and ${mid}Aut({Delta}(G)){mid}$ is square-free if and only if ${mid}G{mid}{leq}3$. Some new graphs that are useful in studying the automorphism group of ${Delta}(G)$ are presented and their main properties are investigated.