Inclusion Criteria:

          -  Age ≥ 18 years at time of informed consent

          -  Body weight > 30 kg

          -  Histologically confirmed diagnosis of GCT (nonseminoma or seminoma in men;
             non-dysgerminomas, dysgerminomas, or germinomas in women or patients with pineal gland
             GCT) at MSKCC of any primary site (includes female GCT and intracranial GCT).

          -  Evidence of measurable disease either by RECIST 1.1 or elevation of serum tumor
             markers (AFP > 15 ng/mL or HCG >2.2 mIU/ml).

          -  Patients must have progressed after at least one prior systemic therapy for GCT and
             meet one of the following criteria:

             a. Patients with evidence of progressive or recurrent GCT after progression prior high
             dose chemotherapy (HDCT) treatment, defined as meeting at least on of the following
             criteria: i. Tumor biopsy of new or growing or unresectable lesions demonstrating
             viable GCT. In the event of an incomplete gross resection where viable GCT is found,
             patients will be considered eligible for this study.

        ii. Consecutive elevated serum tumor markers (HCG or AFP) that are increasing. Increase of
        an elevated LDH alone does not constitute progressive disease.

        iii. Development of new or enlarging lesions in the the setting of persistently elevated
        HCG or AFP, even if the HCG and AFP are not continuing to rise.

        b. Patients deemed not to be a candidate for or benefit from potentially curative HDCT or
        other curative treatment options defined as follows: i. Patients with inadequate renal
        function for HDCT. ii. Patients who have had 3 or more lines of prior chemotherapy as this
        patient population has historically not benefitted from HDCT.

        iii. Patients with late relapse (relapse > 2 years after last therapy) as this patient
        population has historically not benefitted from HDCT.

        iv. Patient with inadequate stem cell collection to move forward with HDCT. v. Patients
        with significant medical or psychosocial comorbidities that are felt to be a
        contraindication to HDCT by the treating investigator.

        NOTE: There is no maximum number of prior treatments allowed "Progression" after prior
        therapy is defined as any one of the following: NOTE: Patients with clinically growing
        "teratoma" (normal declining tumor markers and radiographic or clinical progression) should
        be considered for surgery. In patients with rising tumor markers as their only evidence of
        disease progression where AFP is <30 or HCG is <15, alternate causes of increased levels of
        these markers should be ruled out. (e.g., hypogonadism by testosterone suppression of LH,
        hepatitis, use of marijuana).

          -  Patients with brain metastases are allowed onto the study as long as patients have
             completed their treatment for brain metastasis, no longer require corticosteroids, and
             are asymptomatic. Subjects with neurological symptoms should undergo a head CT scan or
             brain MRI to exclude brain metastasis, at the discretion of the treating physician.

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0-2

          -  Adequate normal organ and marrow function as defined below:

               -  Hemoglobin ≥ 9.0 g/dL

               -  Absolute neutrophil count (ANC) ≥ 1.0 x 109/L (> 1000 per mm3)

               -  Platelet count ≥ 100 x 109/L (>100,000 per mm3)

               -  Serum bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (≤ 3 x
                  institutional ULN in patient's with Gilbert's syndrome)

               -  AST (SGOT)/ALT (SGPT) ≤ 2.5 x institutional ULN unless liver metastases are
                  present, in which case it must be ≤ 5x ULN

               -  Calculated creatinine clearance >30 mL/min by the Cockcroft-Gault formula
                  (Cockcroft and Gault 1976) or by 24-hour urine collection for determination of
                  creatinine clearance

          -  Evidence of post-menopausal status or negative urinary or serum pregnancy test for
             female pre-menopausal patients. Women will be considered post-menopausal if they have
             been amenorrheic for 12 months without an alternative medical cause. The following
             age-specific requirements apply.

               -  Women < 50 years of age would be considered post-menopausal if they have been
                  amenorrheic for 12 months or more following cessation of exogenous hormonal
                  treatment and if they have luteinizing hormone and follicle-stimulating hormone
                  levels in the post-menopausal range for the institution or underwent surgical
                  sterilization (bilateral oophorectomy or hysterectomy).

               -  Women >/= 50 years of age would be considered post-menopausal if they have been
                  amenorrheic for 12 months or more following cessation of all exogenous hormonal
                  treatments, had radiation-induced menopause with last menses > 1 year ago, had
                  chemotherapy-induced menopause with last menses >1 year ago, or underwent
                  surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or
                  hysterectomy).

          -  A positive serum pregnancy test must be confirmed by a pelvic US since some NSGCT may
             secrete beta-hCG and cause a false positive pregnancy. A pelvic US does not need to be
             repeated with each cycle unless the treating physician thinks it is necessary to do
             so.

          -  Female patients of reproductive potential, defined as not surgical sterile (i.e.,
             bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or
             post-menopausal (see above for definition) and non-sterilized males who are sexually
             active with a female partner of reproductive potential must be willing to adhere to
             the following restrictions:

               -  Females of childbearing potential who are sexually active with a non-sterilized
                  male partner must agree to use at least 1 highly effective method of
                  contraception from the time of screening until 180 days after the last dose of
                  durvalumab + tremelimumab combination therapy or 90 days after the last dose of
                  durvalumab monotherapy. Cessation of birth control after this point should be
                  discussed with a responsible physician. Periodic abstinence, the rhythm method,
                  and the withdrawal method are not acceptable methods of birth control. It is
                  strongly recommended that non-sterilized male partners of a female patient must
                  use male condom plus spermicide throughout this period Not engaging in sexual
                  activity for the total duration of the drug treatment and the drug washout period
                  is an acceptable practice.

               -  Non-sterilized males who are sexually active with a female partner of
                  childbearing potential must use a male condom plus spermicide from screening
                  through 180 days after receipt of the final dose of durvalumab + tremelimumab
                  combination therapy or 90 days after receipt of the final dose of durvalumab
                  monotherapy. Periodic abstinence, the rhythm method, and the withdrawal method
                  are not acceptable methods of birth control. Not engaging in sexual activity is
                  an acceptable practice. Male patients should refrain from sperm donation
                  throughout this period. It is strongly recommended that female partners (of
                  childbearing potential) of male patients to also use a highly effective method of
                  contraception throughout this period.

          -  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.

        Exclusion Criteria:

          -  Involvement in the planning and/or conduct of the study (applies to both AstraZeneca
             staff and/or staff at the study site)

          -  Previous enrolment in the present study

          -  Participation in another clinical study with an investigational product during the
             last 14 days

          -  Mean QT interval corrected for heart rate (QTc) ≥ 470 ms calculated from 3
             electrocardiograms (ECGs) using Fredericia's Correction

          -  Any previous treatment with a PD-1 or PD-L1 inhibitor, including durvalumab or an
             anti-CTLA-4, including tremelimumab

          -  Receipt of the last dose of anti-cancer therapy (chemotherapy, immunotherapy,
             endocrine therapy, targeted therapy, biologic therapy, tumor embolization, monoclonal
             antibodies, other investigational agent) ≤ 14 days prior to the first dose of study
             drug

          -  Major surgery within 28 days of starting study treatment. There is no minimum time
             requirement for minor procedures such as biopsy or vascular access placement.

          -  Radiation within 14 days of starting study treatment

          -  Current or prior use of immunosuppressive medication within 28 days before the first
             dose of durvalumab or tremelimumab. The following are exceptions to this criterion:

               -  Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intra
                  articular injection)

               -  Systemic corticosteroids at physiologic doses not to exceed10 mg/day of
                  prednisone or its equivalent

               -  Steroids as premedication for hypersensitivity reactions (e.g., CT scan
                  premedication)

               -  A temporary period of steroids for different indications, at the discretion of
                  the principal investigator (e.g., chronic obstructive pulmonary disease,
                  radiation, nausea, etc)

          -  Any unresolved toxicity (>CTCAE grade 2) from previous anti-cancer therapy. Subjects
             with irreversible toxicity that is not reasonably expected to be exacerbated by the
             investigational product may be included (e.g., alopecia, hearing loss, peripheral
             neuropathy).

          -  History of pulmonary fibrosis by imaging or biopsy (including secondary to bleomycin),
             pneumonitis (including drug induced) requiring steroids ≥ 3 weeks, organizing
             pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or
             evidence of active pneumonitis on screening chest computed tomography (CT) scan with
             associated symptoms.

          -  Active or prior documented autoimmune or inflammatory disorders (including
             inflammatory bowel disease [e.g., colitis or Crohn's disease], diverticulitis, celiac
             disease, systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome
             (granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis,
             hypophysitis, uveitis, etc)). The following are exceptions to this criterion:

               -  Patients with vitiligo or alopecia

               -  Patients with hypothyroidism (eg, following Hashimoto syndrome) stable on hormone
                  replacement

               -  Any chronic skin condition that does not require systemic therapy

               -  Patients without active autoimmune disease in the last 5 years may be included
                  but only after consultation with the study physician

               -  Patients with diverticulosis

               -  Patients with celiac disease controlled by diet alone

          -  History of primary immunodeficiency

          -  History of allogeneic organ transplant

          -  History of hypersensitivity to durvalumab, tremelimumab or any excipient

          -  Uncontrolled intercurrent illness including, but not limited to, on-going or active
             infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable
             angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active
             bleeding diatheses, evidence of 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

          -  Active infection including tuberculosis (clinical evaluation that includes clinical
             history, physical examination and radiographic findings, and TB testing in line with
             local practice), hepatitis B (known positive HBV surface antigen (HBsAg) result),
             hepatitis C, or human immunodeficiency virus (positive HIV 1/2 antibodies). Patients
             with a past or resolved HBV infection (defined as the presence of hepatitis B core
             antibody [anti-HBc] and absence of HBsAg) are eligible. Patients positive for
             hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative
             for HCV RNA.

          -  Known history of previous clinical diagnosis of tuberculosis

          -  History of leptomeningeal carcinomatosis

          -  Receipt of live attenuated vaccination within 30 days prior to study entry or within
             30 days of receiving durvalumab or tremelimumab. Inactivated vaccines, such as the
             injectable influenza vaccine, are permitted.

          -  Any condition that, in the opinion of the investigator, would interfere with
             evaluation of study treatment or interpretation of patient safety or study results

          -  Malignancies other than the disease under study within 5 years prior to Cycle 1, Day
             1, with the exception of those with a negligible risk of metastasis or death and with
             expected curative outcome (such as adequately treated carcinoma in situ of the cervix,
             basal or squamous cell skin cancer, localized prostate cancer treated surgically with
             curative intent, or ductal carcinoma in situ treated surgically with curative intent)
             or undergoing active surveillance per standard-of-care management (e.g. prostate
             cancer with Gleason score 6, and prostate-specific antigen (PSA) 10 mg/mL, etc).

          -  Patients should agree to not donate blood while participating in this study or for at
             least 90 days following the last infusion of durvalumab or tremelimumab

          -  Female patients who are pregnant or breastfeeding