This is a revised and expanded edition of Kac's original introduction to algebraic aspects of conformal field theory, which was published by the AMS in 1996. The volume serves as an introduction to algebraic aspects of conformal field theory, which in the past 15 years revealed a variety of unusual mathematical notions. Vertex algebra theory provides an effective tool to study them in a unified way. In the book, a mathematician encounters new algebraic structures that originated from Einstein's special relativity postulate and Heisenberg's uncertainty principle. A physicist will find familiar notions presented in a more rigorous and systematic way, possibly leading to a better understanding of foundations of quantum physics. This revised edition is based on courses given by the author at MIT and at Rome University in spring 1997. New material is added, including the foundations of a rapidly growing area of algebraic conformal theory. Also, in some places the exposition has been significantly simplified. Readership Graduate students, research mathematicians and physicists working in mathematical aspects of quantum field theory. Reviews "Very good introductional book on vertex algebras."  Zentralblatt MATH "Essential reading for anyone trying to learn about vertex algebras ... well worth buying for experts."  Bulletin of the London Mathematical Society Table of Contents  Preface
 Preface to the second edition
 Wightman axioms and vertex algebras
 Calculus of formal distributions
 Local fields
 Structure theory of vertex algebras
 Examples of vertex algebras and their applications
 Bibliography
 Index
