Abstract
In these lecture notes we give a ``historical'' introduction to topological gauge theories. Our main aim is to clearly explain the origin of the Hamiltonian which forms the basis of Witten's construction of topological gauge theory. We show how this Hamiltonian arises from Witten's formulation of Morse theory as applied by Floer to the infinite dimensional space of gauge connections, with the Chern--Simons functional as the appropriate Morse function(al). We therefore discuss the De Rham cohomology, Hodge theory, Morse theory, Floer homology, Witten's construction of the Lagrangian for topological gauge theory, the subsequent BRST formulation of topological quantum field theory and finally Witten's construction of the Donaldson polynomials.