Dark circles can be caused by several factors and because of that, some dark circles may never vanish.
Dark circles caused by lack of sleep will naturally go away once you have gotten the sleep necessary. Unfortunately, the majority of dark circle complaints are due more to hyperpigmentation of the skin. Dark circles also appear as your skin thins a bit and the underlying blood vessels become more noticeable.
If the darkness is not the result from lack of sleep than there is nothing you can do to permanently get rid of them. Highly hyped "fade creams" can not fade the color of the blood vessels nor can adequately fade hyperpigmentation. Nor can Vitamin K or Horsechestnut (2 other widely advertised remedies).The only thing that can be done is to camoflauge the darkness with a good under-eye concealer and a few make up tips.
If you are a wizard with makeup you can start off using a yellow cream stick to neutralize the darkness. But most don't know how to use yellow, green, and lavendar bases as corrective makeup. So let's skip that and go on to the basic.
When using a concealer you should try to match your skin tone or never go more than one shade lighter or you'll have the "racoon look". Concealers come in a variety: stick, cream, liquid...and can be applied with a brush, sponge tip applicator, or finger tips. If using fingertips, use the ring finger to dab on as that is the weakest finger of your hand and will be the most gentle around the delicate tissue of the eye. Whatever method you use to apply, make sure the color goes up to the bottom lashes and blends in well at the corner of the eye. After applying concealer you can cover with your regular foundation if you so chose. Or if you prefer the natural look just use the concealer, lightly set with transclucent face powder, and you are done.
Practically every major makeup line makes some form of concealer. You can find cheap ones as well as expensive ones. I actually prefer to use a "cream to powder" foundation for a more complete coverage. And for the maximum coverage of everything from dark circle to tatoos to birthmarks, nothing beats
Dermablend.