In this expository review we discuss various aspects of gauge theory. While
the focus is on mathematics, wherever possible we make contact with theoretical
high energy physics. Particular emphasis is placed on instantons and monopoles,
which admit physical interpretation, and yield interesting and nontrivial
mathematics. We give a clear and essentially self-contained exposition of the
mathematical structure of the Seiberg-Witten monopole equations. Other topics
include Donaldson's theorem on moduli spaces of monopoles, compactification of
spaces of ASD connections, The Abelian monopole equations, and Abelian Higgs
vortices.