Let X be a projective scheme. Moduli spaces of semistable coherent sheaves on X are very well-studied, and their construction via (reductive) Geometric Invariant Theory (GIT) is classical. About unstable sheaves, however, little is known, and moduli spaces parameterising them have only been constructed in a few ad hoc examples.

First, I will outline a general framework for GIT-moduli problems, called 'Moduli Spaces of Unstable Objects'. The basic idea is that any moduli problem posed using classical GIT can broken up into manageable pieces, such that each piece has a moduli space: thus one constructs spaces parameterising all objects, not just semistable ones. The key technical tool is a generalisation of GIT to non-reductive groups, which is due to Berczi-Doran-Hawes-Kirwan.

I will then show how to apply these methods to construct moduli spaces of unstable coherent sheaves on X, of any fixed Harder-Narasimhan type. I will focus in particular on the sheaves of HN length 2, and also report on joint work with G.Berczi, F.Kirwan, and V.Hoskins, on sheaves of arbitrary HN length.