The field of vertex operator algebras is an active area of research and plays an integral role in infinite-dimensional Lie theory, string theory, and conformal field theory, and other subdisciplines of mathematics and physics. This book begins with a careful presentation of the theoretical foundations of vertex operator algebras and their modules, and then proceeds to a range of applications. The text features new, original results and a fresh perspective on the important works of many researchers; in particular, it provides a detailed treatment of the concept of a "representation'' of a vertex (operator) algebra. Requiring only a familiarity with basic algebra, this broad, self-contained treatment of the core topics in vertex algebras will appeal to graduate students and researchers in both mathematics and physics.

1 Introduction.- 1.1 Motivation.- 1.2 Example of a vertex operator.- 1.3 The notion of vertex operator algebra.- 1.4 Simplification of the definition.- 1.5 Representations and modules.- 1.6 Construction of families of examples.- 1.7 Some further developments.- 2 Formal Calculus.- 2.1 Formal series and the formal delta function.- 2.2 Derivations and the formal Taylor Theorem.- 2.3 Expansions of zero and applications.- 3 Vertex Operator Algebras: The Axiomatic Basics.- 3.1 Definitions and some fundamental properties.- 3.2 Commutativity properties.- 3.3 Associativity properties.- 3.4 The Jacobi identity from commutativity and associativity.- 3.5 The Jacobi identity from commutativity.- 3.6 The Jacobi identity from skew symmetry and associativity.- 3.7 S3-symmetry of the Jacobi identity.- 3.8 The iterate formula and normal-ordered products.- 3.9 Further elementary notions.- 3.10 Weak nilpotence and nilpotence.- 3.11 Centralizers and the center.- 3.12 Direct product and tensor product vertex algebras.- 4 Modules.- 4.1 Definition and some consequences.- 4.2 Commutativity properties.- 4.3 Associativity properties.- 4.4 The Jacobi identity as a consequence of associativity and commutativity properties.- 4.5 Further elementary notions.- 4.6 Tensor product modules for tensor product vertex algebras.- 4.7 Vacuum-like vectors.- 4.8 Adjoining a module to a vertex algebra.- 5 Representations of Vertex Algebras and the Construction of Vertex Algebras and Modules.- 5.1 Weak vertex operators.- 5.2 The action of weak vertex operators on the space of weak vertex operators.- 5.3 The canonical weak vertex algebra ?(W) and the equivalence between modules and representations.- 5.4 Subalgebras of ?(W).- 5.5 Local subalgebras and vertex subalgebras of ?(W).- 5.6 Vertex subalgebras of ?(W)associated with the Virasoro algebra.- 5.7 General construction theorems for vertex algebras and modules.- 6 Construction of Families of Vertex Operator Algebras and Modules.- 6.1 Vertex operator algebras and modules associated to the Virasoro algebra.- 6.2 Vertex operator algebras and modules associated to affine Lie algebras.- 6.3 Vertex operator algebras and modules associated to Heisenberg algebras.- 6.4 Vertex operator algebras and modules associated to even lattices-the setting.- 6.5 Vertex operator algebras and modules associated to even lattices-the main results.- 6.6 Classification of the irreducible L?(?, O)-modules for g finite-dimensional simple and ? a positive integer.- References.