The max-coloring problem is a natural generalization of the classical vertex coloring problem. The input is a vertex-weighted graph. The objective is to produce a proper coloring such that the overall weight of color classes is minimized, where the weight of each class is defined to be the maximum weight of vertices in that class. Max-coloring has received significant attention over the last decade. Approximation algorithms and hardness results are now known for a number of graph classes in both the offline and online setting. The objective of this chapter is to survey the algorithmic state of the art for the max-coloring problem.