  A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is
a critical point for the L2-norm of the Weyl curvature. When the Riemannian
4-manifold in question is a Kaehler surface, we provide a rough classification
of solutions, followed by detailed results regarding each case in the
classification. The most mysterious case prominently involves 3-dimensional CR
manifolds.
