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For dirty, finger-printed DVDs: dishwashing liquid (mild, for hand washing) seems to work fine with warm water.
Big scratches, I dunno. I talked with our Public Library a few months ago. Their Ken Burns Civil War set had two unplayable discs with ugly circular scratches. The guy at the desk said they would "polish" the discs - but he couldn't explain the actual technique.
Sure enough, a couple of weeks later, I borrowed the set again. The scratches were still visible, but played fine. I will have to talk to someone in their mends dep't to find out what they do.
Video stores should have an answer, too, I suppose.