Superconformal symmetry and superstring compactification

Various topics in conformal field theory and the theory of Kac-Moody algebras are presented. In particular, the Goddard-Kent-Olive construction is used to derive various conformal and superconformal theories, including a large class of N = 2 models recently discovered by Kazama and Suzuki, The relationship between compactification of extra dimensions and the description of internal degrees of freedom by a conformal field theory is discussed. Various approaches to compactification based on exactly soluble conformal field theories, including Gepner's proposal for using the N = 2 minimal models, are sketched. Recent progress in understanding N = 2 models and Calabi-Yau spaces using mathematical techniques of singularity theory is described. It is argued that a classical solution could be a useful first approximation to a quantum ground state even though it is known that string theory is strongly coupled and the perturbation expansion diverges