Subjects with MPM will undergo surgical mediastinal lymph node biopsy (cervical mediastinoscopy) and simultaneous surgical biopsy of the pleural tumor by thoracoscopy, at which time tumor tissue (at least 2 g) and peripheral blood will be collected for the study. These procedures are performed as standard of care in the treatment of these subjects. The subject will be randomized. Three days to three weeks after the biopsy, subjects will be randomly treated with either MEDI-4736 (15 mg/kg once intravenously) or MEDI-4736 (1500 mg once intravenously) plus tremelimumab (75 mg once intravenously) or a control group in a randomized controlled study design. There will be two treatment arms (MEDI4736 only and combination MEDI4736+tremelimumab) and one untreated arm (control). Randomization, stratified by receiving previous chemotherapy or not, will be performed and will help to minimize patient selection biases between three arms. Subjects under 30 kg will be treated with weight-based dosing for both MEDI4736 and Tremelimumab combination therapy. These patients are excluded from fixed based dosing to limit endotoxin exposure from the drug preparations. One to six weeks after the infusion, subjects will undergo resectional surgery, including extrapleural pneumonectomy (EPP) or pleurectomy/decortication (P/D), at which time the tumor will be removed (typically 200-1000 g) and obtained for study. Four patients that do not undergo treatment with MEDI-4736 or tremelimumab will be included as controls. Blood will be obtained after the induction of general anesthesia for both the thoracoscopy procedure and the EPP or P/D resectional procedure, as is routinely done in these procedures. The sixth rib will be obtained at the time of the resection. After the removal of the tumor, standard protocol includes intraoperative heated chemotherapy using a lavage of intracavitary cisplatin in the presence of conserved renal function (Sugarbaker et al., 2013, 2014; Richards et al., 2006).