We want to tell a ghost story. Why?
At the end of the twentieth century it has become clear that quantum field theories describe an essential part of our world. Whether there exists a more fundamental substructure, or rather a superstructure in ten or eleven dimensions like string theories or M-theory, remains to be seen. In any case quantum field theory, in particular quantum gauge theories, will remain the basis of particle physics, and, on the theoretical side, the gauge structure must emerge from any more fundamental theory like string theory, for example. Therefore a mathematically precise formulation of those gauge theories is highly desirable. While the mathematical tools for classical field theories are readily at hand in the form of partial differential equations, the quantum counterpart of functional differential equations is not. The corresponding integral formulation exists for the classical theories in terms of Feynman path integrals. The construction of quantum field theories by means of functional integrals, on the other hand, could not until now be carried out in a rigorous manner for the physically interesting theories, although widely used in the literature . This seems to suggest that the mathematical nature of quantum field theory is rather in the dark. Fortunately, the situation is not so bad.