In this final chapter, we introduce an area of graph theory referred to as extremal graph theory. Of the many problems encountered in this area, a common one asks for the minimum size of a graph G of a given order which guarantees that G contains a certain subgraph or possesses a certain property. While a result of this type due to Paul Tura´n is often considered the beginning of extremal graph theory, there is a host of problems that may be considered as belonging to this area, including problems dealing with the existence of regular graphs of a given degree and a given girth as well as several that lie in a well-known area of graph theory called Ramsey theory.