Introduction by I. R. Shafarevich Chapter 1. Riemann Surfaces 1. Basic Notions 1.1. Complex Chart; Complex Coordinates 1.2. Complex Analytic Atlas 1.3. Complex Analytic Manifolds 1.4. Mappings of Complex Manifolds 1.5. Dimension of a Complex Manifold 1.6. Riemann Surfaces 1.7. Differentiable Manifolds 2. Mappings of Riemann Surfaces 2.1. Nonconstant Mappings of Riemann Surfaces are Discrete 2.2. Meromorphic Functions on a Pdemann Surface 2.3. Meromorphic Functions with Prescribed Behaviour at Poles 2.4. Multiplicity of a Mapping; Order of a Function 2.5. Topological Properties of Mappings of Riemann Surfaces 2.6. Divisors on Riemann Surfaces 2.7. Finite Mappings of Riemann Surfaces 2.8. Unramified Coverings of Pdemann Surfaces 2.9. The Universal Covering 2.10. Continuation of Mappings 2.11. The Riemann Surface of an Algebraic Function 3. Topology of Riemann Surfaces 3.1. Orientability 3.2. Triangulability 3.3. Development; Topological Genus 3.4. Structure of the Fundamental Group 3.5. The Euler Characteristic 3.6. The Hurwitz Formulae 3.7. Homology and Cohomology; Betti Numbers 3.8. Intersection Product; Poincare Duality 4. Calculus on Riemann Surfaces 4.1. Tangent Vectors; Differentiations 4.2. Differential Forms 4.3. Exterior Differentiations; de Rham Cohomology 4.4. Kahler and Riemann Metrics 4.5. Integration of Exterior Differentials; Green's Formula 4.6. Periods; de Rham Isomorphism 4.7. Holomorphic Differentials; Geometric Genus;Riemann's Bilinear Relations 4.8. Meromorphic Differentials; Canonical Divisors 4.9. Meromorphic Differentials with Prescribed Behaviour at Poles; Residues 4.10. Periods of Meromorphic Differentials 4.11. Harmonic Differentials 4.12. Hilbert Space of Differentials; Harmonic Projection 4.13. Hodge Decomposition 4.14. Existence of Meromorphic Differentials and Functions 4.15. Dirichlet's Principle 5. Classification of Riemann Surfaces 5.1. Canonical Regions 5.2. Uniformization 5.3. Types of Riemann Surfaces 5.4. Automorphisms of Canonical Regions 5.5. Riemann Surfaces of Elliptic Type 5.6. Riemann Surfaces of Parabolic Type 5.7. Riemann Surfaces of Hyperbolic Type 5.8. Automorphic Forms; Poincare Series 5.9. Quotient Riemann Surfaces; the Absolute Invariant 5.10. Moduli of Riemann Surfaces 6. Algebraic Nature of Compact Riemann Surfaces 6.1. Function Spaces and Mappings Associated with Divisors 6.2. Riemann-Roch Formula; Reciprocity Law for Differentials of the First and Second Kind 6.3. Applications of the Riemann-Roch Formula to Problems of Existence of Meromorphic Functions and Differentials . 6.4. Compact Riemann Surfaces are Projective 6.5. Algebraic Nature of Projective Models;Arithmetic Riemann Surfaces 6.6. Models of Riemann Surfaces of Genus 1 Chapter 2. Algebraic Curves 1. Basic Notions 1.1. Algebraic Varieties; Zariski Topology 1.2. Regular Functions and Mappings 1.3. The Image of a Projective Variety is Closed 1.4. Irreducibility; Dimension 1.5. Algebraic Curves 1.6. Singular and Nonsingular Points on Varieties 1.7. Rational Functions, Mappings and Varieties 1.8. Differentials 1.9. Comparison Theorems 1.10. Lefschetz Principle 2. Riemann-Roch Formula 2.1. Multiplicity of a Mapping; Ramification 2.2. Divisors 2.3. Intersection of Plane Curves 2.4. The Hurwitz Formulae 2.5. Function Spaces and Spaces of Differentials Associated with Divisors 2.6. Comparison Theorems (Continued) 2.7. Riemann-Roch Formula 2.8. Approaches to the Proof 2.9. First Applications 2.10. Riemann Count 3. Geometry of Projective Curves 3.1. Linear Systems 3.2. Mappings of Curves into Pn 3.3. Generic Hyperplane Sections 3.4. Geometrical Interpretation of the Riemann-Roch Formula 3.5. Clifford's Inequality 3.6. Castelnuovo's Inequality 3.7. Space Curves 3.8. Projective Normality 3.9. The Ideal of a Curve; Intersections of Quadrics 3.10. Complete Intersections 3.11. The Simplest Singularities of Curves 3.12. The Clebsch Formula 3.13. Dual Curves 3.14. Pliicker Formula for the Class 3.15. Correspondence of Branches; Dual Formulae Chapter 3. Jacobians and Abelian Varieties 1. Abelian Varieties 1.1. Algebraic Groups 1.2. Abelian Varieties 1.3. Algebraic Complex Tori; Polarized Tori 1.4. Theta Function and Riemann Theta Divisor 1.5. Principally Polarized Abelian Varieties 1.6. Points of Finite Order on Abelian Varieties 1.7. Elliptic Curves 2. Jacobians of Curves and of Riemann Surfaces 2.1. Principal Divisors on Riemann Surfaces 2.2. Inversion Problem 2.3. Picard Group 2.4. Picard Varieties and their Universal Property . 2.5. Polarization Divisor of the Jacobian of a Curve;Poincare Formulae 2.6. Jacobian of a Curve of Genus 1 References