This book is devoted to two special tools of mathematics: orthogonal polynomials and continued fractions, on the background of historic efforts by Euler (the famous paper “Continued fractions, observation” is included in the book as an appendix). The range of themes covered is very wide including Markoff’s theorem on the Lagrange spectrum, Abel’s theorem on integration in finite terms and Chebyshev’s theory of orthogonal polynomials. Five chapters are devoted to continued fractions (Real numbers, Algebra, Analysis, Euler and Euler’s influence). The sixth chapter studies P fractions (constructed by polynomials). The last two chapters contain a study of orthogonal polynomials and orthogonal polynomials on the unit circle. There are exercises at the end of every chapter.