Mathematical Surveys and Monographs 88. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3674-9/pbk). xiv, 400 p. $ 69.00 (2004).

This is a substantially rewritten and expended second edition of [{\it E. Frenkel, D. Ben-Zvi}, Vertex algebras and algebraic curves. Mathematical Surveys and Monographs. 88. Providence, RI: American Mathematical Society (AMS). (2001;

Zbl 0981.17022)]. The principal changes with respect to the first edition are the following: Throughout the book the authors have dropped the requirement for a vertex algebra to be $\Bbb{Z}$-graded with finite-dimensional graded components. The exposition of associativity and operator product expansions is changed. A more detailed discussion of the Lie algebra $U(V)$, attached to a vertex operator algebra $V$, is included as a separate section. Further, the authors define a topological associative algebra $\tilde{U}(V)$ and prove an equivalence between the category of $V$-modules and the category of smooth $\tilde{U}(V)$-modules. In a separate section the authors introduce twisted modules associated to a vertex algebras equipped with an automorphism of finite order. A direct algebraic proof of coordinate-independence of the connection on the vertex algebra bundle is given. The authors also add new example of chiral algebras, which do not arise from vertex algebras, and explain how to attached to modules and twisted modules over vertex algebras certain modules over the corresponding chiral algebras. Finally, a new chapter on factorization algebras, which provide a purely geometric reformulation of the definition of vertex algebras, is added. In this chapter the authors also discuss chiral Hecke algebras and the geometric Langlands conjecture.