  Gauge theory, which is the basis of all particle physics, is itself based on
a few fundamental concepts, the consequences of which are often as beautiful as
they are deep. In this short lecture course I shall try to give an introduction
to these concepts, both from the physical and mathematical points of view. Then
I shall show how these considerations lead to a nonabelian generalization of
the well-known electric--magnetic duality in electromagnetism. I shall end by
sketching some of the many consequences in quantum field theory that this
duality engenders in particle physics.
  These are notes from a lecture course given in the Summer School on {\em
Geometric Methods in Quantum Field Theory}, Villa de Leyva, Colombia, July
1999, as well as a series of graduate lectures given in Oxford in Trinity Term
of 1999 and 2000.
