I General Formalism. - 1. Rational Continued Functions. - 2. The Continued Fraction of a Real Number. - 3. Equivalent Numbers. - 4. Intermediate Convergents. - II Asymptotic Approximations. - 1. Distribution of the Convergents. - 2. Numbers of Constant Type. - 3. Asymptotic Approximations. - 4. Relation with Continued Fractions. - III Estimates of Averaging Sums. - 1. The Sum of the Remainders. - 2. The Sum of the Reciprocals. - 3. Quadratic Exponential Sums. - 4. Sums with More General Functions. - IV Quadratic Irrationalities. - 1. Quadratic Numbers and Periodicity. - 2. Units and Continued Fractions. - 3. The Basic Asymptotic Estimate. - V The Exponential Function. - 1. Some Continued Functions. - 2. The Continued Fraction for e. - 3. The Basic Asymptotic Estimate. - Appendix A Some Computations in Diophantine Approximations. - Appendix B Continued Fractions for Some Algebraic Numbers. - Appendix C Addendum to Continued Fractions for Some Algebraic Numbers. Language: English