  We continue our study of the noncommutative algebraic and differential
geometry of a particular class of deformations of toric varieties, focusing on
aspects pertinent to the construction and enumeration of noncommutative
instantons on these varieties. We develop a noncommutative version of twistor
theory, which introduces a new example of a noncommutative four-sphere. We
develop a braided version of the ADHM construction and show that it
parametrizes a certain moduli space of framed torsion free sheaves on a
noncommutative projective plane. We use these constructions to explicitly build
instanton gauge bundles with canonical connections on the noncommutative
four-sphere that satisfy appropriate anti-selfduality equations. We construct
projective moduli spaces for the torsion free sheaves and demonstrate that they
are smooth. We define equivariant partition functions of these moduli spaces,
finding that they coincide with the usual instanton partition functions for
supersymmetric gauge theories on C^2.
