The manuscript that follows was written fifteen years ago. On balance, though, conformal field theory has evolved less quickly than I expected, and to my mind the difficulties which kept me from finishing the paper are still not altogether elucidated.

My aim when I began the work was fairly narrow. I was not trying to motivate the study of conformal field theory: I simply wanted to justify my proposed definition, on the one hand by showing that it did encode the usual structure of local field operators and their vacuum expectation values, and on the other hand by checking that all the known examples of conformal theories did fit the definition. As far as the first task is concerned, the crucial part of the paper is §9, where local fields are defined and studied. It was the second task that held me up. The known theories are

1. the σ-model of a torus, or ‘free bosons compactified on a torus’,

2. free fermions,

3. the Wess–Zumino–Witten theory for a compact Lie group,

4. theories obtained from WZW theories by the ‘coset’ construction of Goddard, Kent, and Olive,

5. theories obtained from the preceding ones by the ‘orbifold’ construction.

(I should stress that this is a list of explicit constructions, not a classification of theories. It ignores supersymmetry, and also what I would now call ‘non-compact’ theories.) The crucial case is the WZW theory, which reduces to the representation theory of loop groups.