In graph theory colorings, especially the coloring of maps, are valid topics that have to be investigated from a mathematical point of view. In order to research the smallest number of colors needed for coloring a map, known as the chromatic number, the most important definitions and basic terms of graph theory are outlined. The previously defined characteristics of graphs are then used to solve the Königsberg bridge problem. Subsequently, planar graphs are defined and characterized in order to be able to discuss the coloring of maps. For this characterization the Kuratowski theorem is used and an application example is shown. Based on this, the different graph colorings (vertex, edge and list coloring) can be defined and analyzed, focusing mainly on the chromatic number. Then, the coloring of maps, which can be drawn as a dual graph, will be analyzed and discussed separately. After proving the weaker 6- and 5- color theorems for planar graphs, the 4-color theorem is introduced. The historical development of the problem is closely examined as well: Kempes attempt to prove the 4-color theorem, the development of its proof and the computer-based proof of Appel and Haken are shown and discussed. An overview of various fields of graph theory, which have developed through working with the 4-color theorem, rounds off this paper, referring to earlier discussed chapters of the planar graphs and graph coloring. The factorization of graphs, Hamilton circles, and Matroids, as fields of the graph theory, however, are only briefly mentioned.