This nice little book is devoted to the early development of approximation theory. It mainly concentrates on the role of Russian mathematicians but it confronts it with the development of approximation theory in the West. It first explains the approximation results of Euler connected with geodesy (he constructed a best possible approximation for a map of the Russian Empire satisfying certain side conditions) and Laplace’s contribution to the determination of the best ellipsoidal approximation of the planets. Then more than a quarter of the book is devoted to Chebyshev and his results and another part of about the same size to Saint Petersburg mathematical school (among his direct successors one finds the names Bessel, Sochocki, Zolotarev, A. A. and V. A. Markov, Lyapunov, Voronoy and others). The list of Western mathematicians who contributed to approximation theory includes Klein, Blichfeldt, Kirchberg, Weierstrass, Runge, Borel, Young, Fréchet, Fejér and Jackson. The author also tries to present a closer look at the personalities of those he mentions; we learn many facts about their lives and their philosophical attitudes to different approximation results. He quite often quotes Russian sources (with translations), which will help most readers gain understanding of their personalities. The last chapter describes the work of Pscheborski and Bernstein and their role in the creation of the Kharkov School. The book will be of interest for historians of mathematics and for specialists in approximation theory.