This study is for patients that have been diagnosed with stage III-IVB Head and Neck cancer. The investigational drug in this study is MEDI4736 (Durvalumab). The purpose of the safety lead-in portion of the study is to determine whether adding the study drug MEDI4736 to radiation is safe in patients with head and neck cancer who cannot take the drug cisplatin. The purpose of the phase II/III portion is to compare any good and bad effects of usual radiation plus the study treatment, MEDI4736 (durvalumab), to the usual therapy of radiation plus cetuximab in patients with head and neck cancer who cannot take the drug cisplatin. Participants in the safety lead-in portion can expect to be in this study for approximately 2.5 years. Participants in the Phase II/III portion can expect to be in the study for up to 32 weeks, and then followed by their study doctor to monitor for side effects every year.