Various treatments. As we age, our eyelid skin thins and the fat surrounding the eyes bulge forward. The most common types of "bags" under the eyes are related to either too much fat bulging forward or too much redundant skin in the lower eyelid. The surgical approach depends on which of these is the culprit and thus a consultation with an ophthalmologist will help you decide how best to proceed.
Surgery. The bags under the eyes are normal changes that occur over time. Often, they arise as the fat in the skin below the bags starts to thin out, causing the lower eyelids to appear "puffier" than the cheeks. Most medical treatments do not work. The best treatment is surgical removal or repositioning of the fat under the eyelid skin. See an oculoplastic surgeon for an evaluation.