Dark circles under the eyes arise for essentially three (3) reasons. 

1) Some are due to extra pigment on the skin under the eye. Many people, particularly those from the warmer parts of Europe and Asia, are naturally predisposed to getting dark pigment around the eyes, and this pigment can be interpreted as "dark circles", especially with ageing. 
We can advise on a range of pigment-reducing strategies to decrease this effect, using either OTC or prescription-only formulations.

2) Some arise because the skin under the eye is very thin, and so the fine blood vessels behind this thin skin show through and give the dark circle a bluish hue. This is commoner in northern Europeans and those with thinner skin. Again, the passage of time doesn't help the situation. 
The approach here has to be to help thicken the skin, and we can help this process along with collagen-promoting eye skin care and certain microdermabrasion and laser techniques.

Many patients manifest a combination of these two mechanisms, and so a combination of treatments is employed.

And many patients have these two mechanisms working simultaneously with the third mechanism...

3) For as we get older, most of us get dark circles as cheek fat sinks with gravity and time.

In youth we have full high cheeks plump with fat, creating a single smooth round curve running from just below the eyelid down to the corner of the mouth. There is no gap between the eyelid and the cheek, and light falling on the lower eyelid and upper cheek casts no shadow.

All this fat sinks with time. We all recognise the cheek folds and jowls that develop where the fat falls, but can forget that this sinking fat can leave an absence, a depression and thus a shadow in the space left empty at the top of the cheek. 
This is how these dark circles arise. Absence of fat at the top of the cheeks, leaving a little "speed dip" below the lower eyelid. Light falling here casts a shadow; seen as a dark circle.

Try this. Grab a mirror and watch as you place your fingers around 3cm below your lower eyelid, and press up. You should be then pushing fat back up to form a continuous curve with your lower eyelid, as it once did naturally.
If this manouvre improves your "dark circles", chances are we can help you.

Carefully placing a temporary filler into this dark circle will fill the void left behind by the sinking cheek fat. A new continuous curve is created, shadows occur no longer, and your dark circles look a lot better.