Phase II MEDI4736 in Combination With Chemotherapy for First-Line Treatment of Unresectable Mesothelioma

Criteria: * Histologically and/or cytologically confirmed malignant pleural mesothelioma. * Unresectable disease (defined as the participant not being a candidate for curative surgery). * Measurable disease, defined as at least 1 lesion (measurable) that can be accurately assessed at baseline by computed tomography (CT) or magnetic resonance imaging (MRI) and is suitable for repeated assessment (modified RECIST for pleural mesothelioma). * Available unstained archived tumor tissue sample in sufficient quantity to allow for analyses. At least fifteen unstained slides or a tumor block (preferred). NOTE: A fine needle aspiration sample is not sufficient to make the patient eligible for enrollment. Given the complexity of mesothelioma pathological diagnosis and that these will be newly diagnosed patients it is expected that they will have a core needle biopsy or surgical tumor biopsy as part of their initial diagnostic work up. * Age ≥ 18 years. * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1. * Ability to understand and willingness to sign Institutional Review Board (IRB)-approved informed consent. * Willing to provide archived tumor tissue and blood samples for research. * Adequate organ function as measured by the following criteria, obtained ≤ 2 weeks prior to registration: * Absolute Neutrophil Count (ANC) ≥ 1500/mm³ * Hemoglobin ˃9.0 g/dL * Platelets ˃100,000/mm³ * Serum creatinine clearance (CL)\>60 mL/min by the Cockcroft-Gault formula or by 24-hour urine collection for determination of creatinine clearance. NOTE: Patients with a creatinine Cl ≥ 45 mL/min however ≤ 60 mL/min may be considered for enrollment provided they fulfill all other eligibility criteria, these subjects will receive pemetrexed carboplatin concurrent with durvalumab during the combination phase of treatment. Patients with a creatinine CL\<45 mL/min should not be enrolled. * Albumin ≥ 2.8 g/dL * Total Bilirubin ≤ 1.5x Upper Limit of Normal (ULN) * Aspartate Aminotransferase (AST)/Alanine Aminotransferase (ALT) ≤ 2.5x ULN (≤ 5x ULN in patients with liver metastases) * Women must either be of non-reproductive potential or must have a negative serum pregnancy test upon study entry. * Women must not be pregnant or breastfeeding. * Patient 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. * Patient must not have involvement in the planning and/or conduct of the study. No previous enrollment in the present study. * Patients may not have participated in another clinical study with an investigational product during the last 4 weeks. * Patients must not have any prior systemic therapy (chemotherapy, immunotherapy, endocrine therapy, targeted therapy, biologic therapy, tumor embolization, monoclonal antibodies, and other investigational agent) for mesothelioma. * No previous treatment with a PD1 or PD-L1 inhibitor, including durvalumab or any other agent targeting immune checkpoints. * Patients must not have non-pleural mesothelioma e.g. mesothelioma arising in peritoneum, tunica vaginalis or any serosal surface other than the pleura. * Patients must not have an active second malignancy other than non-melanoma skin cancer or cervical carcinoma in situ. * Patients must not have mean QT interval corrected for heart rate (QTc) ≥470 ms calculated from 3 electrocardiograms (ECGs) using Frediricia's Correction. * Patients must not have symptomatic or uncontrolled brain metastases requiring concurrent treatment, inclusive of but not limited to surgery, radiation and/or corticosteroids (prednisone \>10 mg or equivalent). Surgery, radiation and/or corticosteroids (any dose \>10 mg prednisone equivalent) must have been completed ≥ 2 weeks prior to registration. * Patients must not have uncontrolled seizures. * Patients must not have 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. Standard steroid premedication given prior to chemotherapy or as prophylaxis for imaging contrast allergy should not be counted for this criterion. * No active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease, diverticulitis with the exception of diverticulosis, celiac disease, irritable bowel disease; Wegner syndrome) within the past 2 years. Subjects with vitiligo, alopecia, Grave's disease, or psoriasis not requiring systemic treatment (within the past 3 years) are not excluded. * No history of primary immunodeficiency. * No history of allogeneic organ transplant. * No history of hypersensitivity to durvalumab, cisplatin, carboplatin, pemetrexed or any of their excipients. * No 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 including any subject known to have psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent. * No active infection including tuberculosis (clinical evaluation including: physical examination findings, radiographic findings, positive PPD test, etc.), hepatitis B (known positive HBV surface antigen \[HBsAg\] result), hepatitis C, or human immunodeficiency virus (positive HIV 1/2 antibodies as defined by a positive ELISA test). 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. HIV testing is not required in absence of clinical suspicion. * No known history of leptomeningeal carcinomatosis. * Patients must not have received live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving durvalumab. * Patients must not have any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results.