The main subjects of this book are continued fractions and p-adic numbers in basic number theory and not the applications in applied mathematics, analysis, or theory of functions, which are only shortly mentioned. For number theory, one of the main questions is which advantage continued fractions or p-adic numbers show in competition with other numbers and methods?After an introduction, basic definitions and the history of continued fractions, finite and infinite continued fractions, collected theorems of continued fractions, equivalent numbers, quadratic irrational numbers and reduced numbers, Farey sequences and additional results to continued fractions, Pell's Equation, p-adic numbers, the algebra of p­ adic numbers, and results are discussed.