We study the vertex coloring problem in classes of graphs defined by finitely many forbidden induced subgraphs. Of our special interest are the classes defined by forbidden induced subgraphs with at most 4 vertices. For all but three classes in this family we show either NP-completeness or polynomial-time solvability of the problem. For the remaining three classes we prove fixed-parameter tractability. Moreover, for two of them we give a 3/2 approximation polynomial algorithm.