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The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures. Vertex operator algebras can be viewed as "complex analogues" of both Lie algebras and associative algebras. They are mathematically precise counterparts of what are known in physics as chiral algebras, and in particular, they are intimately related to string theory and conformal field theory. Dong and Lepowsky present a generalization of the theory of vertex operator algebras in a systematic way, in three successive stages, all of which involve one-dimensional braid group representations intrinsically in the algebraic structure: First, their notion of "generalized vertex operator algebra" incorporates such structures as Z-algebras, parafermion algebras, and vertex operator superalgebras. Important examples are based on a general construction that they cau "relative vertex operators." Next, what they term "generalized vertex algebras' encompass in addition the algebras of vertex operators associated with rational lattices. Finally, the most general of the three notions, that of 'abelian intertwining algebra," also iuun-iinates the theory of intertwining operators for certain classes of vertex operator algebras. The monograph is written in an accessible and self-contained way, with detailed proofs and with many examples interwoven through the axiomatic treatment as motivation and applications. It win be useful for research mathematicians and theoretical physicists working in such fields as representation theory and algebraic structures and will provide the basis for a number of graduate courses and seminars on these and related topics.