Quantum Field Theories on Algebraic Curves

In this talk the main features of the operator formalism for the $b-c$ systems on general algebraic curves developed in refs. [1-2] are reviewed. The first part of the talk is an introduction to the language of algebraic curves. Some explicit techniques for the construction of meromorphic tensors are explained. The second part is dedicated to the discussion of the $b-c$ systems. Some new results concerning the concrete representation of the basic operator algebra of the $b-c$ systems and the calculation of divisors on algebraic curves have also been included.