The imminent return to the Moon by multiple space agencies and commercial companies has spurred astronomers to reconsider the Moon as a site for ambitious telescopes. There are several concepts that use the rare properties of special sites on the Moon to undertake astronomical observations not possible elsewhere. These include the well-known radio quiet zone of the lunar farside, the coldest cold traps for far-IR telescopes, and large permanently shadowed regions for cryogenic gravitational wave detectors. For each application there are surprisingly few of these “sites of extraordinary scientific importance.” A first inventory of them is given here. Because these sites are easily degraded by other activities, they will need protection against other uses if the discovery potential they enable is to be kept for humanity. Initial policy approaches to achieve this protection when involving multiple parties are outlined.