The Eyes of Oblivion had a variety of powers. Using hypnosis and an Eye, Professor Gryffen and K9 were able to enter a shared dreamstate (even though K9 was normally unable to dream). When held by the Bodach, the Eye could be used shoot energy beams. Finally, when the two Eyes were set within the eye-sockets of an obelisk, a force field was created and the Bodach could send the whole world to sleep.
The Eyes of Oblivion were set within a Celtic obelisk. With the Eyes, the Bodach planned to send the whole world to sleep, allowing them to feed on their dreams. This plan somehow failed, and the obelisk was buried outside London. In 2050, the obelisk was uncovered and one of the Eyes removed and sent to Professor Gryffen for study. The Bodach tried to get it back, possessing June Turner to do so. Though the Bodach was successful, Jorjie Turner was able to wake up June, releasing her from the Bodach's control and allowing her to take back the Eye. (TV: Dream-Eaters)