The number one cause of circles under the eyes is loss of volume under the eyes (hollowness). We actually lose fat and this is one place where the loose fat here shields what lies underneath: blood vessels (purple) and muscle (reddish brown) which appear to be a brownish grey when viewed through the thin skin of the lower eyelid. The number one remedy for this is injectable filler called Restylane. This works because is puts a layer of clear gel over the darker structure and gently rises up the hollow, improving the transparency and making the skin bright and reflective. Another way to help this is by using a firming cream under the eye such as SkinCeuticals A.G.E. Eye Complex and plumpers and brighteners such as those contained in Colorescience Eye Restoration Kit.
Occasionally under eye circles are caused by loose skin or by pigmentation in the skin. “Loose skin” is when there is too much skin and it gets folded over and causes the skin to look darker. You can tell if your skin is too loose by trying to spread it out with your fingers. If it looks lighter when you spread it then it is loose. The remedy for this would be the same as for under eye wrinkles.
Having allergies or eczema causes another type of pigmentation under the eyes, due to frequent rubbing of their eyes. This leads to skin thickening, which makes the skin appear darker. Treating the underlying allergies is the only effective deal with this.