  This work proves certain general orbifold compactness results for spaces of
Riemannian metrics, generalizing earlier results along these lines for Einstein
metrics or metrics with bounded Ricci curvature. This is then applied to prove
such compactness for spaces of Bach-flat (for example half-conformally flat)
metrics on 4-manifolds, and related results for metrics which are critical
points of other natural Riemannian functionals on the space of metrics.
