  Here we survey questions and results on the Hodge theory of hyperkaehler
quotients, motivated by certain S-duality considerations in string theory. The
problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures
on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons,
magnetic monopoles on R^3 and Higgs bundles on a Riemann surface. Several of
these spaces and their hyperkaehler metrics were constructed by Nigel Hitchin
and his collaborators.
