Periorbital swelling is a medical term for swelling around the eyes which is a temporary condition. This is different from bags which form under the eyes which is a natural part of the aging process. Periorbital swelling could occur in one or both eyes and it can be a gradual symptom which is considered to be chronic or it can happen suddenly, otherwise known as acute. There are various causes of swelling around the eyes, including allergies, but they all result from an excess build up of fluids.