  We obtain a compactness result for various classes of Riemannian metrics in
dimension four; in particular our method applies to anti-self-dual metrics,
Kahler metrics with constant scalar curvature, and metrics with harmonic
curvature. With certain geometric assumptions, the moduli space can be
compactified by adding metrics with orbifold singularities. Similar results
were obtained previously for Einstein metrics, but our analysis differs
substantially from the Einstein case in that we do not assume any pointwise
Ricci curvature bound.
