In order to clean and restore your headlights, you will need the following materials: razor blades, tape, liquid rubbing compound, and a power polisher.
Tape the area surrounding the headlight. To remove the yellowing film, scrape it with a razor blade. This is an alternative to using sandpaper: sandpaper will take much longer. Scrape it until yellow flakes stop coming off. When white flakes alone are coming off, you are done with this step. It may take approximately 7 minutes per headlight.
Rub liquid polishing compound on the headlight. You might want to place a tarp on the rest of your car, to protect it from splatter. Next, use your power polisher. Polish it until it is clear and shiny.