  Using quantum field-theoretic arguments, Witten has established a relation
between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds.
In this survey article, we describe the program to prove this relation using a
moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson
moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1)
monopoles. We provide an overview of some of our transversality and Uhlenbeck
compactness results for PU(2) monopoles, along with some of our calculations of
Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief
outline of issues concerning the gluing theory, focussing on some of the
analytical difficulties that are particular to PU(2) monopoles, and its
application to the PU(2) monopole program to prove the relation between
Donaldson and Seiberg-Witten invariants.
