This book is the second edition of a book with the same title, but in Bulgarian, published in 1963. Nikola Obrechkoff (1896 – 1963) was a world renownеd specialist in the field of the geometry of polynomials. To him are due outstanding generalizations of the Descartes rule, Budan – Fourier theorem, and of several Laguerre’s theorems. In his last publication On some algebraic covariants and the zeros of polynomials he proposed a very promising new approach to the classicial item of the localization of the zeros of polynomials. Unfortunately, this publication, along with the book Zeros of polynomials still remained unknown to the world mathematical community. This edition was supported by the Institute of Mathematics and Informatics of Bulgarian Academy of Sciences, the first director of which academician N. Obrechkoff was from 1951 till 1963. The publishers believe that after 40 years of its first edition, this book still may be useful for the researches in the field of the geometry of polynomials as well as in the numerical analysis in general.