  These notes give an introduction to Geometric Invariant Theory and symplectic
reduction, with lots of pictures and simple examples. We describe their
applications to moduli of bundles and varieties, and their infinite dimensional
analogues in gauge theory and the theory of special metrics on algebraic
varieties. Donaldson's "quantisation" link between the infinite and finite
dimensional situations is described, as are surprisingly strong connections
between the bundle and variety cases.
