I will first review basic results about invariants of actions of algebraic group, especially reductive ones (finite generation of rings of invariants, separation by invariants, linearization). Then I will discuss closure of orbits, surjectivity properties of quotients and the notions of good and geometric quotients, and the Hilbert-Mumford numerical criterion for stability.

In the last week, I will explain how the theory from the first part is worked out for Hilbert points of pluricanonical curves and what this is good for. I will explain what a moduli space is, using the moduli space $M_g$ of smooth curves of genus $g$ as a model, and why finding compactifications of moduli spaces is important. Then I will sketch how the moduli space $\overline{M}_g$ of stable curves is constructed as a GIT quotient and how variants of this construction give rise both to log minimal models arising in the Hassett-Keel program and to alternate compactifications of $M_g$. These variants call for new techniques that will be the subject of my Colloquium talk.

The initial lectures will be largely self-contained; the later ones will use some black boxes, notably the construction of the Hilbert scheme.