Inclusion Criteria: - STEP 1: NEOADJUVANT - 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 have epithelioid or biphasic histology (sarcomatoid histology is excluded); histologic diagnosis and typing of mesothelioma requires at least a core needle biopsy or surgical biopsy of the pleura via thoracoscopy and small thoracotomy; cytology only will not be regarded as sufficient for the diagnosis - Patient must have computed tomography (CT) chest/abdomen/pelvis with contrast or fludeoxyglucose F-18 (FDG)-positron emission tomography (PET)/CT scan performed within 28 days prior to step 1 registration - Patients must have non-measurable or measurable disease documented by CT or magnetic resonance imaging (MRI); the CT from a combined PET/CT may be used to document only non-measurable disease unless it is of diagnostic quality; measurable disease must be assessed within 28 days prior to step 1 registration; non-measurable disease must be assessed within 42 days prior to step 1 registration; all disease must be assessed and documented on the RECIST 1.1 and modified RECIST baseline tumor assessment form - Patient must have undergone extended surgical staging including mediastinoscopy or endobronchial ultrasound; at minimum, samples must be obtained from the mediastinal stations 4R, 7 (subcarinal), and 4L; this surgical staging must be performed within 42 days prior to step 1 registration; patient must be T1-3 and N0-N2 (single station) - Patient must undergo video-assisted thoracoscopic surgery and diagnostic laparoscopy within 28 days prior to step 1 registration to rule out peritoneal disease spread - Patient must have consultation with a surgeon within 21 days prior to step 1 registration; the surgeon must confirm that the patient''s disease is resectable by pleurectomy decortication (P/D) or extrapleural pneumonectomy (EPP) and that the patient is an appropriate candidate for the surgical procedures - Patient must not have had prior immunotherapy or chemotherapy for malignant pleural mesothelioma - Patient must have Zubrod performance status 0 or 1 documented within 28 days prior to step 1 registration - Patients requiring hearing aids or reporting hearing loss must have audiogram performed within 28 days prior to step 1 registration - Patient must have not had any major surgery or radiation within 28 days prior to step 1 registration; diagnostic thoracotomies and laparoscopies are not considered major surgeries - Patients must not have any anticancer therapy or investigational agent within 28 days prior to step 1 registration - Absolute neutrophil count (ANC) >= 1,500/mcl (documented within 28 days prior to step 1 registration) - Hemoglobin >= 9 g/dl (documented within 28 days prior to step 1 registration) - Platelets >= 100,000/mcl (documented within 28 days prior to step 1 registration) - Creatinine =< 1.5 x upper limit of normal (ULN) (documented within 28 days prior to step 1 registration) - Creatinine clearance >= 45 ml/min (documented within 28 days prior to step 1 registration) - Total bilirubin =< 1.5 x upper limit of normal (ULN) (within 28 days prior to step 1 registration) - Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 3 x ULN (within 28 days prior to step 1 registration) - No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for three years - Patients must not be pregnant or nursing due to the potential teratogenic side effects of the protocol treatment; women of reproductive potential and men must have agreed to use an effective contraceptive method for the duration of study treatment and for 5 months (150 days) after the last dose of atezolizumab; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures - Patient must NOT have a history of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins - Patient must NOT have a known hypersensitivity or allergy to biopharmaceuticals produced in Chinese hamster ovary cells or any component of the atezolizumab formulation - Patients must not have severe infections within 28 days prior to step 1 registration, including but not limited to hospitalization for complications of infection, bacteremia, or severe pneumonia - Patients must not have active autoimmune disease that has required systemic treatment in past two years (i.e., with use of disease modifying agents, corticosteroids or immunosuppressive drugs); replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment; autoimmune diseases include, but are not limited to, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener''s granulomatosis, Sjogren''s syndrome, Bell''s palsy, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis; this protocol includes an immunotherapy agent which can precipitate known autoimmune diseases - Patients must not have undergone prior allogeneic bone marrow transplantation or prior solid organ transplantation - Patient must not have active tuberculosis - Patient must not have history of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis; this protocol includes an immunotherapy agent which can precipitate known pneumonitis - Patient must not have active (chronic or acute) hepatitis B virus (HBV) infection as evidenced by testing performed within 28 days prior to registration; patients with past or resolved HBV infection are eligible; active HBV is defined as having a positive hepatitis B surface antigen (HBsAg) test; past or resolved HBV is defined as having a negative HBsAG test and a positive total hepatitis B core antibody (HBcAb) test; patient must not have active hepatitis C virus (HCV) infection as evidenced by testing performed within 28 days prior to registration; active HCV is defined as having a positive HCV antibody test followed by a positive HCV RNA test - Patient must NOT have a known positive test for human immunodeficiency virus (HIV); patients do not need to be screened for HIV; patients with HIV are excluded due to a potential incompetent immune system and need for medications that could interfere with the treatment and immunotherapy - Patient must not have significant cardiovascular disease, such as New York Heart Association cardiac disease (class II or greater), myocardial infarction within 3 months prior to initiation of treatment, unstable arrhythmias, or unstable angina given the higher risks associated with surgical resection - Patient must not receive live, attenuated influenza vaccine within 4 weeks prior to registration or at any time during the study and until 5 months after the last dose of atezolizumab - Patient must be willing to have tissue specimens submitted for translational medicine studies - Patient must be offered the opportunity to participate in tissue and blood banking for future studies - Patient must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines - As a part of the Oncology Patient Enrollment Network (OPEN) registration process, the treating institution''s identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system - STEP 2: SURGERY - Patient must have a CT of chest/abdomen with contrast or FDG-PET/CT scan within 28 days prior to step 2 registration; patients must not have evidence of progression per RECIST 1.1 or modified RECIST for pleural tumors - Patients planning to receive EPP must also be evaluated for appropriateness of radiation therapy (RT) by a radiation oncologist within 14 days prior to step 2 registration - Patients must have a Zubrod performance status of 0-1 documented within 28 days prior to step 2 registration - Patients must have postoperative predicted forced expiratory volume in 1 second (FEV1) > 35% prior to surgery obtained within 28 days prior to step 2 registration; pulmonary function tests to ascertain these values must be obtained within 28 days prior to Step 2 registration - Patients must have postoperative predicted carbon monoxide diffusing capability (DLCO) > 35% prior to surgery obtained within 28 days prior to step 2 registration; pulmonary function tests to ascertain these values must be obtained within 28 days prior to Step 2 registration - Patient must have received at least two cycles of triplet neoadjuvant therapy (all three drugs) during step 1 - Patient must be registered to step 2 no less than 21 days and no more than 90 days after the end of their final cycle of neoadjuvant therapy - STEP 3: MAINTENANCE - Patient must have received either P/D or EPP and must have recovered from all effects of surgery with adequate wound healing; patients who received radiation therapy (RT) must be registered to step 3 within 90 days after discontinuing RT; patients who did not receive RT must be registered to step 3 within 90 days after surgery - Patient must have a CT of chest/abdomen/pelvis with contrast or FDG-PET/CT scan within 28 days prior to step 3 registration; patient must not have evidence of progression per RECIST 1.1 or modified RECIST for pleural tumors - Patient may have discontinued RT early due to toxicity or other reasons - Patients must have a Zubrod performance status of 0-1 documented within 28 days prior to step 3 registration - ANC > 1,500/mcl (documented within 28 days prior to step 3 registration) - Hemoglobin > 9 g/dl (documented within 28 days prior to step 3 registration) - Platelets > 100,000/mcl (documented within 28 days prior to step 3 registration) - Creatinine < 1.5 x ULN (documented within 28 days prior to step 3 registration) - Total bilirubin =< 1.5 x upper limit of normal (ULN) (within 28 days prior to step 3 registration) - Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 3 x ULN (within 28 days prior to step 3 registration)