This book on continued fractions is devoted to certain selected topics in the analytic theory, with particular emphasis on those aspects that deal with rational approximations to functions and with numerical applications and computations. This text contains a tremendous mass of valuable formulas in continued fraction theory. Due to this fact, it can be considered as a useful reference manual for such formulas as well as a text on methods for research in analysis and in computational work.In the first chapter of the present work a short exposition of the analytic theory of continued fractions is given. Problems in the arithmetic theory of continued fractions are not considered in this book.The second chapter is devoted to the continued fraction expansion (by the method of Lagrange) of some well known functions. All expansions given in this chapter are special cases of a general expansion derived at the beginning of the chapter.In the third chapter there is a short consideration of further methods for deriving rational function approximations to functions, leading to a series of approximation formulae for computing certain well known functions.In the fourth chapter are considered the generalized continued fractions proposed by Euler. Examples are quoted showing the possibility of further generalizations of continued fractions which permit the approximate solution of algebraic equations of arbitrary degree