  We prove generic regularity and Uhlenbeck-type compactification theorems for
the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the
usual way (by applying Sard theorem to a smooth parameterized moduli space),
since the parameterized moduli space can be a priori singular. We explain why,
using the standard order 0-perturbations, one cannot get the smoothness of the
parameterized moduli space. (The same difficulty arises in the case of
ASD-Spin^c moduli spaces, so the definition of the Spin^c-polynomial invariants
should be revised.) We show that the singular locus of the parameterized moduli
space is contained in a closed subspace which is (in a certain sense) of
infinite codimension, and we apply Sard theorem to the complement of this
subspace. Our approach to prove Uhlenbeck compactness follows closely the
strategy developed in the instanton case by Donaldson and Kronheimer. Similar
results, with different methods, were obtained by Feehan - Leness and Feehan.
