The lectures concentrate on highlights in combinatorial (Chapters II and III) and number theoretical (Chapter IV) extremal theory, particularly on the solution of famous problems that remained open for many decades. However, the organization of the lectures in six chapters does not follow the historical developments or the connections between ideas in several cases. With the specified auxiliary results in Chapter I on probability theory, graph theory, etc., all chapters can be read and taught independently of one another. In addition to the 16 lectures organized in six chapters of the main part of the book, there is supplementary material for most of them in the appendix. In particular, there are applications and further exercises, research problems, conjectures, and even research programs.