IC/95/339 hep-th/9511038 New Results in Topological Field Theory and Abelian Gauge Theory

These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological field theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main hope is to wet the readers appetite, so that he or she will wish to read the original works and perhaps to enter this field.