There are many different options- most of which have undergone significant advancement in the last decade or so. The best solution must be customized and individualized for each patient.
For someone with mild volume loss below the eyes (tear trough), hyaluronic acid fillers such as Restylane or Juvederm are excellent. In the occasional patient with excessive lower lid fat pads, a transconj. blepharoplasty with incision behind the lid may be a great option to reduce the bags. In others who have excess skin laxity and hollows below the eyes, redraping the existing fat, supporting the muscle and trimming the skin conservatively (blepharoplasty with fat transposition) is often the way to go. When the bags are lower in the midface (festoons), other options are more appropriate.
The good news is that there are many state-of-the-art options ranging from non-surgical to elaborate surgical procedures.
An Oculoplastic Surgeon (see ASOPRS) will very likely be highly experienced in the full range of possibilities for bags under the eyes.
Mark Lucarelli MD FACS