Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed thyroid carcinoma of
             follicular origin (including papillary, follicular, hurthle cell or poorly
             differentiated subtypes and their respective variants).

          -  Diagnosis of recurrent and/or metastatic thyroid cancer

          -  At least one RAI-avid lesion identified on the most recent radioiodine scan (a
             diagnostic, post-therapy, or post-ablation scan) OR at least one lesion on the most
             recent FDG PET scan with an SUV max of 10 or less. (Both RAI-sensitive and
             RAI-refractory patients are eligible if at least one tumor with RAI avidity of any
             degree can be identified within one of these parameters.)

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional
             techniques or as ≥ 10 mm with CT scan, MRI, or calipers by clinical exam. See Section
             11 for the evaluation of measurable disease. Tumors in previously irradiated fields
             may be considered measureable if there is evidence of tumor progression after
             radiation treatment.

          -  ECOG Performance Status (PS) 0 or 1. (or Karnofsky ≥60%)

          -  Age ≥ 18 years at time of study entry

          -  Adequate normal organ and marrow function as defined below:

               -  Hemoglobin ≥ 9.0 g/dL

               -  Absolute neutrophil count (ANC) ≥ 1.5 x 10^9/L (> 1500 per mm^3)

               -  Platelet count ≥ 100 x 10^9/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
                  hyperbilirubinemia 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: ≥60 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.

          -  Patients must agree to undergo two research biopsies of (a) malignant lesion(s).
             Biopsies do not need to be done if the investigator or person performing the biopsy
             judges there is no tumor accessible for biopsy, the only accessible tumor must be used
             for RECIST measurement, or the biopsy poses too great a risk to the patient. If the
             patient has only one RECIST measureable target lesion for response assessment,
             research biopsies must not be performed on that target lesion.

          -  Availability of archival tumor tissue from the thyroid cancer primary or metastasis (a
             tissue block or a minimum of 30 unstained slides would be required. Patients with less
             archival tissue available may still be eligible for the study after discussion with
             the MSK Principal Investigator.)

        Exclusion Criteria:

          -  131I therapy < 6 months prior to initiation of therapy on this protocol. A diagnostic
             study using < 10 mCi of 131I is not considered 131I therapy.

          -  Any previous treatment with a PD1 or PD-L1 inhibitor, including durvalumab.

          -  History of pneumonitis.

          -  External beam radiation therapy < 4 weeks prior to initiation of therapy on this
             protocol.

          -  Chemotherapy, immunotherapy, targeted therapy, monoclonal antibodies, tumor
             embolization, or other investigational agent within 28 days prior to the first dose of
             study drug.

          -  Current or prior use of immunosuppressive medication within 28 days before the first
             dose of durvalumab, 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 an equivalent corticosteroid.

          -  Any unresolved toxicity CTCAE grade ≥ 2 from previous anti-cancer therapy. Exceptions
             include hearing loss, peripheral neuropathy, and alopecia.

          -  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 a history of autoimmune thyroid disease are not excluded. Subjects with vitiligo
             or psoriasis not requiring systemic treatment (within the past 2 years) are not
             excluded.

          -  Active or prior documented inflammatory bowel disease (e.g., Crohn's disease,
             ulcerative colitis).

          -  History of primary immunodeficiency.

          -  History of allogeneic organ transplant.

          -  History of hypersensitivity to durvalumab or any excipient.

          -  History of hypersensitivity to thyrotropin alpha (Thyrogen).

          -  Patients unable to follow a low iodine diet or requiring medication with high content
             in iodide (amiodarone).

          -  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 acute or chronic hepatitis B, hepatitis C 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 active tuberculosis.

          -  Symptomatic brain metasteses, leptomeningeal carcinomatosis, or spinal cord
             compression (treated metastatic brain, leptomeningeal carcinomatosis, or spinal cord
             compression are allowed). Note: Patients must be off steroids used for brain
             metasteses, leptomeningeal carcinomatosis, or spinal cord compression.

          -  Receipt of live attenuated vaccination within 30 days prior to study entry or within
             30 days of receiving durvalumab.

          -  Female subjects who are pregnant, breast-feeding or male or female patients of
             reproductive potential who are not employing an effective method of birth control.

          -  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.