  The D-instanton partition function is a fascinating quantity because in the
presence of N D3-branes, and in a certain decoupling limit, it reduces to the
functional integral of N=4 U(N) supersymmetric gauge theory for multi-instanton
solutions. We study this quantity as a function of non-commutativity in the
D3-brane theory, VEVs corresponding to separating the D3-branes and alpha'.
Explicit calculations are presented in the one-instanton sector with arbitrary
N, and in the large-N limit for all instanton charge. We find that for general
instanton charge, the matrix theory admits a nilpotent fermionic symmetry and
that the action is Q-exact. Consequently the partition function localizes on
the minima of the matrix theory action. This allows us to prove some general
properties of these integrals. In the non-commutative theory, the contributions
come from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integral
of the resolved instanton moduli space. Separating the D3-branes leads to
additional localizations on products of abelian instanton moduli spaces. In the
commutative theory, there are additional contributions from the ``Coulomb
Branch'' associated to the small instanton singularities of the instanton
moduli space. We also argue that both non-commutativity and alpha'-corrections
play a similar role in suppressing the contributions from these singularities.
Finally we elucidate the relation between the partition function and the Euler
characteristic of the instanton moduli space.
