Dip a cotton swab in rubbing alcohol or window cleaning solution.
Take the cotton swap and wipe the disc from the inner edge of the disc to the outer edge of the disc. This will effectively remove smudges and dust from the disc's surface.
Take some Brasso and put it on the areas of the CD that are scratched.
Rub the Brasso on the CD with a cloth using a circular motion. For deeper scratches, rub the CD firmly to improve the performance. You may need to apply Brasso to the CD more than once depending on the severity of the scratches. However, be careful how much you use, as the Brasso will remove some of the CD's plastic layer along with the scratches. Once the scratch is gone, or nearly gone, rinse the CD with water and dry with a clean cloth in a circular motion. You have now successfully removed scratches and smudges from a CD.