This work gives an exhaustive exposition of the main facts about the Bernstein polynomials and discusses some of their applications in Analysis. The author writes in the Preface to this second edition ``After the trigonometric integrals, Bernstein polynomials are the most important and interesting concrete operators on a space of continuous functions. Since the appearance of the first edition of this book [in 1953], the interest in this subject has continued. In an appendix we have summed up a few of the most important papers that have appeared since.''