  We survey some features of equivariant instanton partition functions of
topological gauge theories on four and six dimensional toric Kahler varieties,
and their geometric and algebraic counterparts in the enumerative problem of
counting holomorphic curves. We discuss the relations of instanton counting to
representations of affine Lie algebras in the four-dimensional case, and to
Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For
resolutions of toric singularities, an algebraic structure induced by a quiver
determines the instanton moduli space through the McKay correspondence and its
generalizations. The correspondence elucidates the realization of gauge theory
partition functions as quasi-modular forms, and reformulates the computation of
noncommutative Donaldson-Thomas invariants in terms of the enumeration of
generalized instantons. New results include a general presentation of the
partition functions on ALE spaces as affine characters, a rigorous treatment of
equivariant partition functions on Hirzebruch surfaces, and a putative
connection between the special McKay correspondence and instanton counting on
Hirzebruch-Jung spaces.
