First of all, remove all the dust present on the face of the disc. You can use warm water for this. Run the water over the damaged disc or clean its face with a piece of soft cloth dampened with warm water.
If there are any stubborn dust patches, gently rub the area with your finger. You may even use any gentle detergent or rubbing alcohol to get rid of the stubborn dust. In case you are using rubbing alcohol, use a small cotton swab.
Let the disc get dry itself. There is no need to use any towel, cloth or sun-dry. Just shaking the water off the disc is enough.
It is the time to play the disc. If the disc still refuses to play properly, use another player. Sometimes, a device does not play one specific disc properly, as it fails to handle the scratches.
If the CD or DVD works on a different player, burn a new disc. The disc burner usually does not face any difficulty in creating a new copy. However, this should be used as last resort.
If you are dealing with serious scratches, first of all identify the area that is most damaged. You may need to use a scratch remover to handle these serious scuffs.
Once the scratches are completely gone, polish your disc by using baking soda toothpaste or Brasso. A fine-grit polishing compounds is also useful in this regard.