A bounded vertex coloring of a graph G is a usual vertex coloring in which each color is used at most k times (where k is a given number). The bounded chromatic number _k (G) of G is the smallest number of colors such that G admits a bounded coloring. Upper and lower bounds on _k (G) are given in terms of k, the number n of vertices, the usual chromatic number (G) and the maximum degree (G) of G. Complexity of bounded vertex coloring is discussed and several classes of graphs for which this problem is polynomially solvable are identified.