The complexity of counting problems

These lectures are intended to cover various aspects of the complexity of counting problems. They are concentrated on the class of functions contained in the class #P, which is the counting counterpart of the language class NP. These notes are a mixture of theory and examples, taken mainly from pure mathematics and combinatories, Because of the apparent difficulty of most problems which arise naturally, there has been considerable effort put into good approximation schemes. This is reflected in these notes.