The Donaldson invariants of a smooth 4-manifold M are subtle probes into the differential topological structure of M. They are defined from the structure of a moduli space of solutions to certain partial differential equations proposed by the physicists (Yang-Mills) to model the interactions between elementary particles. Ed Witten found they could also be regarded as expectation values for certain observables in a Topological Quantum Field Theory. His insight into the physics behind this TQFT led him to conjecture that the information contained in the Donaldson invariants can also be retrieved from a much simpler set of equations (Seiberg-Witten). The impact of the conjecture, although still not fully proved, has been spectacular. The rather modest objective of this lecture is to provide the background required to understand what the conjecture says.