This book contains selected chapters from the monograph “Introduction into the Theory of Uniform Approximation of Functions by Polynomials” (published by the first author in 1977 in Russian). The second author has complemented them with some chapters from his book “Approximation by Polynomials and Traces of Functions Continuous on a Segment” (published in 1992) and has made the final redaction of the text, since the first author passed away in 1998. On 480 pages, the authors present classical material like Tschebyshev theory and Weierstrass theorems including their further development. In other chapters, the authors treat various topics, e.g. smoothness of functions and direct and inverse theorems on the approximation of periodic functions. The last chapter is devoted to approximation by polynomials. The book also contains 40 pages of references and a two page subject index. At the end of each chapter there are interesting additional remarks. Explanations and proofs are clear and detailed. The authors try to trace the origins of theorems, which is of interest in the case of Russian authors (many sources were published in Russian). The book could be of interest for all who work in approximation theory and related fields; it should not be overlooked by university libraries.