The aim of this talk is to give an introduction to Geometric Invariant Theory (GIT) for reductive groups over the complex numbers. Roughly speaking, GIT is concerned with constructing quotients of group actions in the category of algebraic varieties. We begin by discussing what properties we should like quotient varieties to possess, highlighting so-called `good' and `geometric' quotients, and then turn to search for these quotients in the case of affine and projective varieties. Here we shall see that the construction runs most smoothly when we assume our group to be reductive (meaning it can be described as the complexification of a maximal compact subgroup). Finally, we hope to say something about the Hilbert-Mumford criterion regarding semi-stability and stability of points, illustrating it by constructing the rough moduli space of elliptic curves.
