The stability of polar oxide surfaces has long been a problematic question in surface science. A bulk terminated polar surface has an infinite surface energy because alternating layers of oppositely charged ions produce a large dipole moment perpendicular to the surface. Such a singularity presents many interesting questions ranging from the fundamental "Can polar oxide surfaces exist?" via the mechanistic "How can they get stabilized?" to the applied "Would they have unique and useful surface and interface properties?". Both theory and experiment have provided several contrary answers to the first two questions, and the last is largely unexplored. In the seventies, the problem was considered closed with consensus between theory and experiment that polar oxide surfaces can not exist but must facet into neutral planes to gain finite surface energy. In the nineties the problem was reopened with experimental discoveries of reconstructed polar oxide surfaces and with theoretical predictions of clean reconstructed surfaces based on the idea of smallest neutral building blocks. At present, there is disagreement between the few proposed and solved polar oxide surface structures, and the reconstruction mechanism is under construction. An additional controversy surrounds the 1x1 structure of polar oxide surfaces. Classical electrostatic approaches predict that such structures can exist only by adsorption of charged species, OH being the currently favored termination, but quantum mechanical approaches predict two dimensional surface metalization of the clean 1x1 surface. I will review the present state of knowledge, with illustrations from our multi-technique experimental studies of the polar MgO and NiO (111) surfaces and their neutral (100) and (110) counterparts. Our data favors the reconstruction mechanism at high temperatures, and the OH adsorption mechanism at lower temperatures.