This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.

Ivan Arzhantsev received his doctoral degree in 1998 from Lomonosov Moscow State University and is a professor in its department of higher algebra. His research areas are algebraic geometry, algebraic groups and invariant theory.

Introduction; 1. Basic concepts; 2. Toric varieties and Gale duality; 3. Cox rings and combinatorics; 4. Selected topics; 5. Surfaces; 6. Arithmetic applications.