CEA involves a neck incision and physical removal of the plaque from the inside of the artery. CAS involves insertion of a catheter or tube into an artery in the groin and then threading the catheter through the arteries of the body to the location of the plaque within the carotid artery in the neck. The stent is then placed to cover the plaque and hold the artery open. Participants will be randomly assigned to undergo either CAS or CEA, and all patients will receive best medical management, which includes treatment with aspirin, treatment of high blood pressure, and treatment of other stroke risk factors. Participants will be followed for up to ten years.