This phase III trial compares durvalumab to the usual approach (patient observation) after surgery for the treatment of patients with early-stage non-small cell lung cancer. Immunotherapy with monoclonal antibodies, such as durvalumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. The usual approach for patients who are not in a study is to closely watch a patient's condition after surgery and to have regular visits with their doctor to watch for signs of the cancer coming back. Usually, patients do not receive further treatment unless the cancer returns. This study will help determine whether this different approach with durvalumab is better, the same, or worse than the usual approach of observation. Giving durvalumab may help patients live longer and prevent early-stage non-small cell lung cancer from coming back as compared to the usual approach.