In this thesis, we present an expanded account of star coloring of graphs, based on an article by Albertson et al. (2004). A proper vertex coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color. A proper vertex coloring of a graph G is called a star coloring if the union of every two color dir=rtl align=right Star coloring of graphs was introduced in 1973 by Grünbaum. In 1983, Coleman and Moré found that the problem of estimation of sparse Hessian matrices can be transformed to the problem of star coloring of graphs. Fertin et al., within two papers in 2001 and 2003, studied star coloring of graphs and obtained the exact values and also upper bounds for the star chromatic number of some known families of graphs. They found bounds for the star chromatic number of a graph in terms of its other parameters, such as treewidth and maximum degree. In 2001, Ne?et?il and Ossana de Mendez, studied star coloring of graphs and obtained some important results. Albertson et al., in 2004, considered star coloring of graphs broadly by new ideas and introduced an extension of proper coloring which contains star coloring as a special case. Wood and P?r, in 2007, found a bound for the star chromatic number of Cartesian product of graphs. In years 2008 and 2009, some bounds for the star chromatic number of planar graphs were obtained by different researchers, while the best bound is still unknown. Lyons, in 2009, found some results on the star chromatic number of join of graphs, and also considered algorithms for computing the star chromatic number of graphs in these families. This thesis contains a survey on the results mentioned above. The contents of chapters are as follows: In chapter one, we study the initial concepts and a brief history of the subject. In chapter two, we see the star chromatic number of some special and famous families of graphs. Also, a coloring which is equivalent to star coloring and is useful to obtain bounds for the star chromatic number of graphs, is introduced. In chapter three, we study bounds which obtained on the star chromatic number of Cartesian product of graphs, and a result for the star coloring of join of graphs. In chapter six, we study an application of star coloring and consider the complexity of the problem of k-star colorability of graphs.