Brief Summary: This phase II trial studies how well pembrolizumab works in treating patients with malignant mesothelioma, a cancer of the linings around the lungs (pleura) or abdomen (peritoneum). Monoclonal antibodies, such as pembrolizumab, work by blocking a protein called programmed cell death 1 (PD-1) which may stimulate an immune response and kill tumor cells.

Detailed Description: PRIMARY OBJECTIVES: I. To determine the objective response rate of patients with malignant mesothelioma treated with pembrolizumab in A) an unselected patient population, as well as B) in a programmed cell death ligand 1 (PD-L1) positive population (should the trial proceed to Part B, and PD-L1 expression correlate with improved efficacy). II. To determine the optimal threshold for PD-L1 expression using the 22C3 antibody based immunohistochemistry (IHC) assay in correlation to tumor response. SECONDARY OBJECTIVES: I. To determine the progression-free survival of patients with malignant mesothelioma in A) an unselected patient population and B) a PD-L1 positive population when treated with pembrolizumab. II. To determine the overall survival of patients with malignant mesothelioma in A) an unselected patient population and B) a PD-L1 positive population when treated with pembrolizumab. III. To determine the disease control rate (complete response \[CR\] + partial response \[PR\] + stable disease \[SD\]) of patients with malignant mesothelioma who are treated with pembrolizumab in A) an unselected patient population and B) a PD-L1 positive population. IV. To determine toxicity in patients with malignant mesothelioma who are treated with pembrolizumab. V. To determine percentage of patients with mesothelioma who have PD-L1 tumor expression, and the distribution of PD-L1 expression (percent positivity of tumor cells/stroma staining). TERTIARY OBJECTIVES: I. To characterize the T-cell inflamed phenotype in mesothelioma patients via presence of cluster of differentiation (CD)8 tumor infiltrating lymphocytes (TILs) and/or use of a gene expression signature (Nanostring). II. To evaluate other immune escape mechanisms including indoleamine-pyrrole 2,3-dioxygenase (IDO) expression, regulatory T cells (Tregs) (forkhead box P3 \[FOXP3\] expression), myeloid-derived suppressor cells (MDSCs) and other checkpoints by immunohistochemistry (or other methods e.g. flow cytometry). III. To determine PD-L1 expression by mass spectrometry and correlate with tumor response, PD-L1 expression by IHC, and the T-cell inflamed phenotype. IV. To determine the immune cell populations present in fresh tumor (when available), via tumor digests and mass spectrometry-based flow cytometric analysis (e.g. using CyTOF) in a multiplex fashion in patients with fresh tumor tissue. V. To characterize the T-cell receptor repertoire of TILs compared to circulating T-cells in mesothelioma patients with available fresh frozen tissue (spectrotyping, T-cell repertoire sequencing \[e.g. using the Adaptive platform\]). OUTLINE: Patients receive pembrolizumab intravenously (IV) over 30 minutes on day 1. Treatment repeats every 21 days for up to 24 months in the absence of disease progression or unacceptable toxicity. Patients may be eligible for up to 1 year of additional pembrolizumab therapy if they progress after stopping pembrolizumab. After completion of study treatment, patients are followed up for 30 days (up to 90 days for serious adverse events), every 8 weeks until patient experiences confirmed disease progression or starts a new anti-cancer therapy, and then every 12 weeks for 3 years.