This book is devoted to the systematic presentation of the theory of algebraic functions from one variable over an arbitrary field of constants. The advantage of the book is a rather successful combination of harmony of presentation, conducted taking into account the latest achievements in abstract algebra, with an abundance of actual material. The last chapter of the book provides an upgraded exposition of the classical theory of algebraic functions from one variable over a field of complex numbers. The book is of interest to mathematical researchers, graduate students and senior students of mathematical faculties working in the field of algebra, algebraic geometry and number theory .. To read it, preliminary information from general field theory is required, and in the last chapter, some information from topology and function theory.