Abstract: In a typical extremal problem one wants to determine maximum cardinality of discrete structure with certain prescribed properties. Probably the earliest such result was obtained 100 years ago by Mantel who computed the maximum number of edges in a triangle-free graph on n vertices. This was generalized by Turan to all complete graphs and became a starting point of Extremal Graph Theory.

In this talk we survey several classical problems and results in this area and present some interesting applications of Extremal Graph Theory to other areas of mathematics. We also describe a recent surprising generalization of Turan's theorem which was motivated by question in Computational Complexity.