Evaluating Immune Therapy, Duravalumab (MEDI4736) With Tremelimumab for Relapsed/Refractory Germ Cell Tumors

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