We study the moduli space M_G(A) of flat G-bundles on an Abelian surface A,
where G is a compact, simple, simply connected, connected Lie
group. Equivalently, M_G(A) is the (coarse) moduli space of s-equivalence
classes of holomorphic semi-stable G_C-bundles with trivial Chern classes.
M_G(A) has the structure of a hyperkahler orbifold. We show that when G is
Sp(n) or SU(n), M_G(A) has a natural hyperkahler desingularization which we
exhibit as a moduli space of G_C-bundles with an altered stability
condition. In this way, we obtain the two known families of hyperkahler
manifolds, the Hilbert scheme of points on a K3 surface and the generalized
Kummer varieties. We show that for G not Sp(n) or SU(n), the moduli space
M_G(A) does *not* admit a hyperkahler resolution.
Inspired by the physicists Vafa and Zaslow, Batyrev and Dais define
``stringy Hodge numbers'' for certain orbifolds. These numbers are
conjectured to agree with the Hodge numbers of a crepant resolution (when
it exists). We compute the stringy Hodge numbers of M_{SU(n)}(A) and
M_{Sp(n)}(A)$ and verify the conjecture in these cases.