Diophantine analysis is devoted to diophantine approximations and diophantine equations. Here the word diophantine means that we are concerned about integral or rational solutions. The exact meaning will become clear in the sequel. In this introductory chapter we shall learn some of the basic principles of diophantine analysis. These are special methods (as Fermat’s method of inﬁnite descent) as well as building bridges to other ﬁelds (e.g., the use of real analysis or geometry for understanding underlying arithmetic structures). In order to present these principles we prove some fundamental historical results.