1. Basic definitions -- 2. The invariant bilinear form and the generalized Casimir operator -- 3. Integrable representations and the Weyl group of a Kac-Moody algebra -- 4. Some properties of generalized Cartan matrices -- 5. Real and imaginary roots -- 6. Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group -- 7. Affine Lie algebras: the realization (case k = 1) -- 8. Affine Lie algebras: the realization (case k = 2 or 3). Application to the classification of finite order automorphisms -- 9. Highest weight modules over the Lie algebra g(A) -- 10. Integrable highest weight modules: the character formula -- 11. Integrable highest weight modules: the weight system, the contravariant Hermitian form and the restriction problem -- 12. Integrable highest weight modules over affine Lie algebras. Application to ?-function identities -- 13. Affine Lie algebras, theta functions and modular forms -- 14. The principal realization of the basic representation. Application to the KdV-type hierarchies of non-linear partial differential equations -- Index of notations and definitions -- References

1. Mathematics
2. Topological groups
3. Lie groups
4. Differential geometry
5. Number theory
6. Topology
7. Combinatorics
8. Physics
9. Mathematics
10. Topological Groups
11. Lie Groups
12. Mathematical Methods in Physics
13. Topology
14. Differential Geometry
15. Number Theory
16. Combinatorics