Pembrolizumab in Recurrent or Metastatic Medullary Thyroid Cancer

Background: Medullary thyroid cancer (MTC) is a tumor of the thyroid gland. Surgery is the only current treatment to cure it. The drug pembrolizumab (MK-3475) is a new type of cancer therapy. It works by allowing the immune system to detect and kill tumor cells. Objective: To test how pembrolizumab affects people with MTC and if it can offer them clinical benefit. Eligibility: People ages 18 and older with MTC Patients who have recurrent or metastatic MTC, for whom surgery is not a curative option Patients with some imaging evidence of MTC Patients with minimal symptoms related to MTC Design: Participants will be screened with: * Medical history * Physical exam * Blood, urine, and heart tests * Computed tomography (CT) scan or magnetic resonance imaging (MRI): They lie in a machine that takes pictures of the body. * Bone scan Participants will be put in a group based on their treatment history: * Group 1 if they have had an immune stimulating cancer vaccine * Group 2 if they have had no vaccine Participants will receive the study drug as a 30-minute intravenous (IV) infusion every 3 weeks. Treatment will continue for up to 2 years as long as they tolerate it and their disease does not get worse. Participants will have physical exams and blood tests on the day of each infusion. They will have CT and bone scans every 3 months. Participants may save biopsies before treatment and after starting treatment. Participants will have a final visit 3-4 weeks after stopping treatment. This will include a physical exam and blood and heart tests. After this study, participants can join a long-term follow-up study.