Coloring is one of the important areas in graph theory which is attended by many authors. A vertex coloring of a graph is a function from vertex set of graph to positive integers. Every vertex coloring on a graph, assigns a positive integer or a color to each vertex under special conditions. In 2005, Chartrand et al. introduced a new vertex coloring for the connected graphs which is called Hamiltonian coloring. This coloring, defines a relation between the colors of every two distinct vertices of graph and the length of a longest paht between them. An important parameter of Hamiltonian coloring is Hamiltonian chromatic number. In this thesis, we study the concept of Hamiltonian coloring and provide the comprehensive survey on the obtained results in this subject. In Chapter 1, we consider elementary definitions of graph theory and the history of emersion of Hamiltonian coloring. In Chapter 2, Hamiltonian coloring as a result of another vertex coloring with name radio coloring, has been studied. In Chapter 3, Hamiltonian chromatic number of some well-known graphs has been proved. In Chapter 4, we study some possible and impossible values of Hamiltonian chromatic number of connected graphs. Moreover, some results about the relation between Hamiltonian chromatic number and the length of a longest cycle in graph, has been considered. In Chapter 5, a sharp bound for Hamiltonian chromatic number of connected graphs with a Hamiltonian connected subgraph without large order has been shown. Hamiltonian connected graphs are the only class of connected graphs with Hamiltonian chromatic number one. Finally, we introduce the concept of critical Hamiltonian connected graphs as a class of Hamiltonian connected graphs with optimal edges and study the properties of these graphs.