The affine Kac-Moody algebra $A_1^{(1)}$ has recently served as a source of new ideas in the representation theory of infinite-dimensional affine Lie algebras. In particular, several years ago it was discovered that $A_1^{(1)}$ and then a general class of affine Lie algebras could be constructed using operators related to the vertex operators of the physicists' string model. This book develops the calculus of vertex operators to solve the problem...