In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most b and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes' weights. These problems generalize the well known max-coloring problems by taking into account the number of available resources (i.e., the cardinality of color classes) in practical applications. In this paper we present complexity results and approximation algorithms for the bounded max-coloring problems on general graphs, bipartite graphs and trees.