The book, addressed to students of physical and mathematical specialties, was written on the basis of lectures given by the authors at the Independent Moscow University .. The first part describes the foundations of the theory of algebraic curves considered as Riemannian surfaces. It is dominated by relatively elementary algebraic and geometrical methods. A novelty for educational literature of this level is the discussion of the relationship of algebraic curves with Galois theory. For the first time in Russian, Ritt's theorems on compositions of polynomials and on commuting polynomials. are presented. In the second part of the book, the initial is already the treatment of the Riemannian surface itself as a complex one-dimensional manifold .. The theorems on topological, holomorphic and hyperbolic uniformization, Poincaré's method of constructing non-permanent meromorphic functions, the great Ponsele theorem. In both parts, the general concepts and results are illustrated by numerous examples. There are many tasks in the book that are designed not only to help the reader better master the material, but also to provide significant additional information.