The purpose of this study is to test the safety of durvalumab (MEDI4736) and tremelimumab in combination with radiation therapy and find out what effects, if any, this combination has on people, and whether it improves overall survival.

Inclusion Criteria:

• Histopathologic confirmation of anaplastic thyroid cancer (or histopathologic report consistent with anaplastic thyroid cancer) at Memorial Sloan Kettering Cancer Center with clinical evidence of metastatic disease not curable by either surgery or radiation therapy

• Age ≥ 18 years at time of study entry

• ECOG Performance Status of 0-2

• Adequate normal organ and marrow function as defined below:

  • Hemoglobin ≥ 9.0 g/dL
  • Absolute neutrophil count (ANC) ≥ 1.5 x 109/L (≥ 1500 per mm^3)
  • Platelet count ≥ 100 x 109/L (≥100,000 per mm^3)
  • Serum bilirubin ≤ 1.5 x institutional upper limit of normal (ULN). This will not apply to subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinaemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology), who will be allowed only in consultation with their physician
  • AST (SGOT)/ALT (SGPT) ≤ 2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be ≤ 5x ULN
  • Serum creatinine CL>40 mL/min by the Cockcroft-Gault formula (Cockcroft and Gault 1976) or by 24-hour urine collection for determination of creatinine clearance:
  • Males:

Creatinine CL (mL/min) = Weight (kg) x (140 - Age) . 72 x serum creatinine (mg/dL)

°Females: Creatinine CL (mL/min) = Weight (kg) x (140 - Age) x 0.85 72 x serum creatinine (mg/dL)

• Female subjects must either be of non-reproductive potential (i.e., post-menopausal by history: ≥51 years old and no menses for ≥ 1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry
• Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up
• Measurable disease as per RECIST 1.1 besides the tumor that is expected to be radiated. See Section 12.2.1 for detailed description. Bony lesions without soft tissue component are not measurable lesions
• Patient must agree to allow 2 separate biopsies of any malignant lesion. Biopsies do not need to be done if:

A. Tumor is not considered accessible by either the investigator or the person performing the biopsy (it is determined the risk is too high due to location near vital organs or too great of a risk of an adverse event).

B. Patient is on anticoagulation and it would be unsafe to temporarily hold the anticoagulation.

C.Consent of the PI not to have a biopsy done.

Exclusion Criteria:

• Any previous treatment with an anti-CTLA4, including tremelimumab or any previous treatment with a PD1 or PDL1 inhibitor
• Receipt of the last dose of any line of anti-cancer therapy (chemotherapy, immunotherapy, targeted therapy, biologic therapy, tumor embolization, monoclonal antibodies, and other investigational agent) 7 days prior to the first dose of study drug and within 6 weeks for nitrosourea or mitomycin C Prior radiation therapy to targets other than the site currently being treated is permitted.
• Mean QT interval corrected for heart rate (QTc) ≥470 ms calculated from a single electrocardiogram or average from 3 electrocardiograms (ECGs) using Frederica's Correction
• Current or prior use of immunosuppressive medication within 28 days before the first dose of durvalumab (MEDI4736) or tremelimumab, with the exceptions of intranasal and inhaled corticosteroids, or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone or any corticosteroid for more than 4 consecutive days
• Any unresolved toxicity (CTCAE grade > 1) from previous anti-cancer systemic therapy unless approved by one of the principal investigators (Drs. Lee or Sherman) Rare exceptions that would not affect the study (e.g., radiaton induced dysphagia) allowed with the consent of either principal investigators (Drs. Lee or Sherman)
• Any prior Grade ≥3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE >Grade 1
• Active or prior documented autoimmune disease within the past 2 years Note: Subjects with vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded. Rare exceptions allowed with the consent of both principal investigators (Drs. Lee and Sherman)
• Active or prior documented inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis).
• Confirmed pneumonitis or interstitial lung disease
• History of primary immunodeficiency
• History of allogeneic organ transplant
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active bleeding diatheses, Any subject known to have evidence of acute or chronic active hepatitis B, hepatitis C (PCR positive) or human immunodeficiency virus (HIV), or Psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent
• Known history of previous clinical diagnosis of active tuberculosis (This does not include a history of being PPD positive)
• History of leptomeningeal carcinomatosis or active brain metastases receiving concurrent treatment inclusive of but not limited to surgery, radiation and/or corticosteroids. Note: Brain metastases that have been treated for anticancer purposes where there has been no MRI evidence of progression for at least 8 weeks after treatment are permitted on study
• Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving durvalumab (MEDI4736) or tremelimumab
• Any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results Subjects with uncontrolled seizures
• Female patients who are pregnant or breast feeding or male or female patients of reproductive potential who are not willing to employ effective birth control from screening to 180 days after the last dose of durvalumab (MEDI4736) + tremelimumab combination therapy or 90 days after the last dose of durvalumab (MEDI4736) monotherapy, whichever is the longer time period
• Prior active malignancy within the previous 3 years except for locally curable cancers such as basal or squamous cell cancer superficial bladder, low risk prostate cancer, breast, or cervix cancer unless it is believed the malignancy will not affect the study outcome