Lie algebras were originally introduced by S. Lie as algebraic structures used for the study of Lie groups. The tangent space of a Lie group at the identity element has the natural structure of a Lie algebra, called by Lie the infinitesimal group. However, Lie algebras also proved to be of interest in their own right. The finite dimensional simple Lie algebras over the complex field were investigated independently by E. Cartan and W. Killing and the classification of such algebras was achieved during the decade 1890–1900. Basic ideas on the structure and representation theory of these Lie algebras were also contributed at a later stage by H. Weyl. Since then the theory of finite dimensional simple Lie algebras has found many and varied applications both in mathematics and in mathematical physics, to the extent that it is now generally regarded as one of the classical branches of mathematics.

In 1967 V. G. Kac and R. V. Moody independently introduced the Lie algebras now known as Kac–Moody algebras. The finite dimensional simple Lie algebras are examples of Kac–Moody algebras; but the theory of Kac–Moody algebras is much broader, including many infinite dimensional examples. The Kac–Moody theory has developed rapidly since its introduction and has also turned out to have applications in many areas of mathematics, including among others group theory, combinatorics, modular forms, differential equations and invariant theory. It has also proved important in mathematical physics, where it has applications to statistical physics, conformal field theory and string theory. The representation theory of affine Kac–Moody algebras has been particularly useful in such applications.