  We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide
an explicit local description, which we use to obtain the complete
classification in the compact case. We give new examples of extremal Kahler
metrics, including Kahler-Einstein metrics and conformally Einstein Kahler
metrics. We also extend some of our results to almost Kahler 4-manifolds,
providing new examples of Ricci-flat almost Kahler metrics which are not
Kahler.
