Patient must have stage I-III malignant pleural mesothelioma that is deemed resectable and must be planning to undergo pleurectomy decortication (P/D) or extrapleural pneumonectomy (EPP) Patient must not have had prior immunotherapy or chemotherapy for malignant pleural mesothelioma Histologically confirmed pleural malignant mesothelioma not eligible for curative surgery Prior chemotherapy for pleural mesothelioma Histologically or cytologically confirmed unresectable or medically inoperable malignant pleural mesothelioma Mesothelioma Mesothelioma. Patients with a histologic diagnosis of malignant pleural mesothelioma (MPM), epithelioid subtype, who in the opinion of the attending thoracic surgeon can receive a macroscopically complete resection of tumor Part B3 only: Participants must have malignant pleural or peritoneal mesothelioma Part B4 only: Participants must have malignant pleural or peritoneal mesothelioma and appropriate candidate for treatment with cisplatin/pemetrexed; no prior systemic chemotherapy COHORT B, GROUP 5: MESOTHELIOMA: Patients must have histologically or cytologically proven diagnosis of malignant mesothelioma; both pleural and peritoneal mesothelioma are allowed Patients with sarcomatoid mesothelioma as mesothelin is not expressed in this type of mesothelioma Patients must have histologically or cytologically malignant mesothelioma confirmed by the National Cancer Institute (NCI) Laboratory of Pathology. Patients with pleural, peritoneal, pericardial or tunica vaginalis mesothelioma are eligible Arm A 14-patients expansion cohort: patients with histologically or cytologically proven chemotherapy naive unresectable malignant pleural or peritoneal mesothelioma Arm B (first stage of phase II of TRC102 and pemetrexed): patients with malignant pleural or peritoneal mesothelioma who had progressed while being treated with or had recurred within 6 months of being treated with pemetrexed and cisplatin or carboplatin frontline; intervening treatment is allowed Pathologic diagnosis of malignant mesothelioma (any primary site is acceptable, any histology is acceptable) Have unresectable malignant mesothelioma (any histology) Patients must have a histologic diagnosis of malignant pleural mesothelioma Patients who have received pleurectomy with decortication (P/D) or EPP for mesothelioma Histologically proven malignant pleural mesothelioma MPM that is considered resectable according to the following criteria:\r\n* Confined to one pleural space\r\n* No chest wall invasion\r\n* No transdiaphragmatic involvement\r\n* No invasion of mediastinal structures Histologically or cytologically confirmed pleural malignant mesothelioma, epithelial or biphasic subtypes Histologically confirmed epithelioid predominantly (> 70%) subtype malignant pleural mesothelioma Patients must have histologically or cytologically confirmed malignant pleural mesothelioma (MPM) Surgically resectable malignant pleural mesothelioma (MPM) with no disease extension beyond the ipsilateral hemithorax Patients must have a pathologically confirmed diagnosis, either at MSKCC or at the participating site, of stage I-III malignant pleural mesothelioma Prior nephrectomy on the contralateral side of malignant pleural mesothelioma (MPM) Measurable disease as defined by RECIST v1.1 or disease-specific criteria for prostate cancer and malignant pleural mesothelioma Patients must have histologically or cytologically confirmed malignant mesothelioma for which they have received pemetrexed in combination with cisplatin as part of chemotherapeutic regimen. Patients must not have received any other chemotherapeutic treatment for malignant mesothelioma other than pemetrexed and a platinum agent such as cisplatin. Patient must have histologically or cytologically confirmed diagnosis of malignant pleural mesothelioma; surgical resection must not be planned Has mesothelioma (pleural, peritoneal, pericardial, tunica vaginalis) of any histology that is relapsed or refractory after treatment with at least one pemetrexed-containing regimen Peritoneal mesothelioma basket \r\n* Refractory or intolerant to platinum and pemetrexed systemic therapy\r\n* Any number of prior therapies Peritoneal mesothelioma basket:\r\n* None Pleural mesothelioma basket:\r\n* None predominantly epithelial (?50% tumor component) pleural or peritoneal mesothelioma Currently have mesothelioma of the sarcomatous type, mixed histologic disease, or have malignant peritoneal mesothelioma Pathologic diagnosis of malignant mesothelioma (any primary site is acceptable) Have unresectable malignant mesothelioma (any histology) Malignant pleural mesothelioma (histologically confirmed epithelial) Sarcomatoid malignant pleural mesothelioma (MPM) histology which is known in the literature to not express mesothelin; biphasic MPM is also excluded Histologically confirmed malignant pleural mesothelioma (MPM) (Epithelioid or biphasic subtype for Phase II patients; epithelioid subtype only for Phase III patients) Histological documentation of malignant pleural mesothelioma (MPM) overexpressing mesothelin Patients must have histologically confirmed diagnosis of unresectable malignant pleural mesothelioma ELIGIBILITY FOR ENROLLMENT/SCREENING (ARMS 1 AND 2): Histopathological documentation of NSCLC or mesothelioma Histologically or cytologically confirmed pleural or peritoneal malignant mesothelioma, epithelial, sarcomatoid, or biphasic subtypes Have measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 for peritoneal mesothelioma, and modified RECIST for pleural mesothelioma Histologically confirmed malignant pleural mesothelioma that is not metastatic or unresectable Histologically or cytologically confirmed malignant pleural or peritoneal mesothelioma (MPM) Patients must have a histologically confirmed diagnosis of epithelioid, sarcomatoid, or mixed-type malignant pleural or peritoneal mesothelioma that is not amenable to surgery Patients must have measurable disease according to the modified Response Evaluation Criteria in Solid Tumors (RECIST) criteria for pleural mesothelioma, or standard RECIST for peritoneal mesothelioma; patients must have adequate tissue sample available for molecular profiling with Memorial Sloan-Kettering (MSK)-IMPACT (archived tissue block or 15-20 unstained slides); patients will sign a separate informed document (Institutional Review Board [IRB] #12-245) to allow this to be performed Histologically documented malignant pleural mesothelioma, epithelial, sarcomatoid or mixed type, not amenable to surgical resection Patients with unresectable, malignant pleural or peritoneal mesothelioma who have progressed on first-line pemetrexed-based chemotherapy Histologically proven malignant pleural or peritoneal mesothelioma of epithelioid or biphasic histology Patients with only one kidney that is ipsilateral to the mesothelioma Histopathologically-confirmed diagnosis of malignant mesothelioma (pleural or peritoneal). Must have disease that has relapsed following at least one prior line of chemotherapy. Mesothelin-positive refractory/recurrent solid tumors, other than malignant pleural mesothelioma (MPM) and pancreatic ductal adenocarcinoma (PDA) (Group 1 only) Assessment by the attending thoracic surgeon that radical pleurectomy can be safely achieved in pts with malignant pleural mesothelioma 6. Have completed baseline quality of life evaluation as assessed by LCSS modified for mesothelioma Pathologic diagnosis of malignant pleural mesothelioma (MPM) confirmed at participating institution Histologically confirmed peritoneal carcinomatosis from appendiceal, colorectal, ovarian, or primary mesothelioma, with no systemic metastases