  We survey and compare recent approaches to the computation of the partition
functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge
theories on ALE spaces based on quiver varieties and the minimal resolution
$X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$, in light
of their recently conjectured duality with two-dimensional coset conformal
field theories. We review and elucidate the rigorous constructions of gauge
theories for a particular family of ALE spaces, using their relation to the
cohomology of moduli spaces of framed torsion free sheaves on a suitable
orbifold compactification of $X_k$. We extend these computations to generic
$\mathcal{N}=2$ superconformal quiver gauge theories, obtaining in these
instances new constraints on fractional instanton charges, a rigorous proof of
the Nekrasov master formula, and new quantizations of Hitchin systems based on
the underlying Seiberg-Witten geometry.
