Riemann surfaces, conformal fields and strings

The role of Riemann surfaces in modern particle physics is discussed. Mathematically, quantum field theories can be defined on these manifolds if they are conformally invariant. Physically, Riemann surfaces provide a model for the world sheet swept out by a propagating relativistic string. Thus Riemann surfaces are the natural setting for conformal field theory, and both these concepts together provide a formulation of string theory. (author)