This work is about extended affine Lie algebras (EALA's) and their root systems. EALA's were introduced by HøeghKrohn and Torresani under the name irreducible quasisimple Lie algebras. The major objective is to develop enough theory to provide a firm foundation for further study of EALA's. The first chapter of the paper is devoted to establishing some basic structure theory. It includes a proof of the fact that, as conjectured by Kac, the invariant symmetric bilinear form on an EALA can be scaled so that its restriction to the real span of the root system is positive semidefinite. The second chapter studies extended affine root systems (EARS) which are an axiomatized version of the root systems arising from EALA's. The concept of a semilattice is used to give a complete description of EARS. In the final chapter, a number of new examples of extended affine Lie algebras are given. The concluding appendix contains an axiomatic characterization of the nonisotropic roots in an EARS in a more general context than the one used in the rest of the paper. Features:  Provides a foundation for the study of an important class of Lie algebras that generalizes the class of affine KacMoody Lie algebras
 Includes material on Lie algebras and on root systems that can be read independently.
Readership Graduate students, research mathematicians and mathematical physicists interested in Lie theory. Table of Contents  Introduction
 The basic structure theory of extended affine Lie algebras
 Semilattices and extended affine root systems
 Examples of extended affine Lie algebras
 Appendix. Axiomatic theory of roots
 References
