When a reductive group G acts on a complex projective variety
X, there exist different methods for finding an open G-invariant subset
of X with a geometric quotient (the 'stable locus'), which is a
quasi-projective variety and has a projective completion X//G. Mumford's
geometric invariant theory (GIT) developed in the 1960s provides one way
to do this, given a lift of the action to an ample line bundle on X,
though with no guarantee that the stable locus is not empty. An
alternative approach due to Kapranov and others in the 1990s is to use
Chow varieties to define a 'Chow quotient' X//G. The aim of this talk is
to review the relationship between these constructions for reductive
groups, and to discuss the situation when G is not reductive.

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.