Atezolizumab and Bevacizumab in Treating Patients With Rare Solid Tumors

Inclusion Criteria: * Signed informed consent form * Ability to comply with the study protocol, in the investigator's judgment * Measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1; previously irradiated lesions can be considered as measurable disease only if progressive disease has been unequivocally documented at that site since radiation; the pleural mesothelioma cohort will require measurable disease according to modified RECIST * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L without granulocyte colony-stimulating factor support, obtained within 14 days prior to initiation of study treatment * Lymphocyte count \>= 0.5 x 10\^9/L, obtained within 14 days prior to initiation of study treatment * Platelet count \>= 100 x 10\^9/L without transfusion, obtained within 14 days prior to initiation of study treatment * White blood cell (WBC) count \>= 2500/ul, obtained within 14 days prior to initiation of study treatment * Hemoglobin \>= 90 g/L (patients may be transfused to meet this criterion), obtained within 14 days prior to initiation of study treatment * Aspartate aminotransferase (AST), alanine aminotransferase (ALT), and alkaline phosphatase (ALP) =\< 2.5 x upper limit of normal (ULN), obtained within 14 days prior to initiation of study treatment, with the following exceptions: patients with documented liver metastases: AST and ALT =\< 5 x ULN; patients with documented liver or bone metastases: alkaline phosphatase (ALP) =\< 5 x ULN * Serum bilirubin 1.5 x ULN, obtained within 14 days prior to initiation of study treatment * Serum creatinine =\< 1.5 x ULN, obtained within 14 days prior to initiation of study treatment * Serum albumin \>= 2.5 g/dL, obtained within 14 days prior to initiation of study treatment * For patients not receiving therapeutic anticoagulation: international normalized ratio (INR) or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN, obtained within 14 days prior to initiation of study treatment * For patients receiving therapeutic anticoagulation: stable anticoagulant regimen * For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of \< 1% per year during the treatment period and for 6 months after the last dose of study treatment * A woman is considered to be of childbearing potential if she is postmenarchal, has not reached a postmenopausal state (\>= 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus); examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices; the reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient; periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception * Appendiceal adenocarcinoma basket * Metastatic appendiceal adenocarcinoma * Not considered candidate for curative surgery * Nasopharyngeal carcinoma basket * Metastatic or locally recurrent disease not amenable to curative intent treatment * Any number of prior therapies, including 0 * Human papilloma virus-associated cancers * Histologically proven squamous carcinoma of the anal canal, penile, vaginal, vulva, or refractory cervical cancer with progression or intolerance to at least one treatment regimen including cisplatin, oxaliplatin or carboplatin will be enrolled; human papilloma virus (HPV) confirmation is not required * Patients must have metastatic disease not amenable to surgical resection * If human immunodeficiency virus (HIV)+ positive, all patients infected with human immunodeficiency virus (HIV) and CD4+ T cell count \> 400 cells/mm\^3 may be eligible for study * Patients co-infected with hepatitis B virus and/or hepatitis C virus may be included in this study provided that their liver function tests remain within the limits listed above; patients must be followed by a hepatologist during the course of this study * Merkel cell carcinoma basket * Metastatic or locally recurrent disease not amenable to curative intent treatment * Prior treatment with CD137 agonists or immune checkpoint blockade therapies, including anti-CTLA-4, anti-PD-1, and anti-PD-L1 therapeutic antibodies * Any number of prior therapies * Neuroendocrine tumors, pancreatic basket * Grade 1 or grade 2 (or described as low grade, intermediate grade, well differentiated, or moderately differentiated) according to reviewing pathologist * Progressive disease over the preceding 12 months * Any number of prior therapies, including 0 * Patients using a somatostatin analogue for symptom control must be on stable doses for 56 days prior to enrollment * Neuroendocrine tumors, extrapancreatic basket * Grade 1 or grade 2 (or described as low grade, intermediate grade, well differentiated, or moderately differentiated; typical or atypical carcinoid if originating in lung) according to reviewing pathologist * Progressive disease over the preceding 12 months * Any number of prior therapies, including 0 * Patients using a somatostatin analogue for symptom control must be on stable doses for 56 days prior to enrollment * Peritoneal mesothelioma basket * Refractory or intolerant to platinum and pemetrexed systemic therapy * Any number of prior therapies * Pleural mesothelioma basket * Metastatic or locally recurrent disease not amenable to curative intent treatment * Refractory to platinum and pemetrexed systemic therapy * Any number of prior therapies Exclusion Criteria: * Treatment for the studied cancer within 28 days prior to initiation of study treatment * Treatment with investigational therapy within 28 days prior to initiation of study treatment * History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins * Known hypersensitivity to biopharmaceutical agents produced in Chinese hamster ovary cells * Known allergy or hypersensitivity to any component of the atezolizumab formulation * Known allergy or hypersensitivity to any component of the bevacizumab formulation * Active or history of autoimmune disease or immune deficiency, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjogren's syndrome, Guillain-Barre syndrome, or multiple sclerosis, with the following exceptions: patients with a history of autoimmune-related hypothyroidism who are on thyroid replacement hormone are eligible for the study; patients with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study; patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible for the study provided all of following conditions are met: * Rash must cover \< 10% of body surface area * Disease is well controlled at baseline and requires only low-potency topical corticosteroids * No occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high potency or oral corticosteroids within the previous 12 months * Prior allogeneic stem cell or solid organ transplantation * History of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on screening chest computed tomography (CT) scan; (history of radiation pneumonitis in the radiation field \[fibrosis\] is permitted) * Positive HIV test at screening (except in cohort 3, HPV-associated cancers) * Except in cohort 3, HPV-associated cancers, active hepatitis B virus (HBV) infection (chronic or acute), defined as having a positive hepatitis B surface antigen (HBsAg) test at screening; patients with a past or resolved HBV infection, defined as having a negative HBsAg test and a positive total hepatitis B core antibody (HBcAb) test and negative HBV deoxyribonucleic acid (DNA) test at screening, are eligible for the study * Except in cohort 3, HPV-associated cancers active hepatitis C virus (HCV) infection, defined as having a positive HCV antibody test followed by a positive HCV RNA test at screening; the HCV RNA test will be performed only for patients who have a positive HCV antibody test * Active tuberculosis * Severe infection within 4 weeks prior to initiation of study treatment, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia * Treatment with therapeutic oral or IV antibiotics within 2 weeks prior to initiation of study treatment; patients receiving prophylactic antibiotics (e.g., to prevent a urinary tract infection or chronic obstructive pulmonary disease exacerbation) are eligible for the study * Significant cardiovascular disease, such as New York Heart Association cardiac disease (class II or greater), myocardial infarction, or cerebrovascular accident within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina * Major surgical procedure other than for diagnosis within 4 weeks prior to initiation of study treatment, or anticipation of need for a major surgical procedure during the course of the study * Treatment with a live, attenuated vaccine within 4 weeks prior to initiation of study treatment, or anticipation of need for such a vaccine during the course of the study, or up to 5 months following the anticipated last dose of atezolizumab * Malignancies other than the disease under study within 5 years prior to cycle 1, day 1, with the exception of those with a negligible risk of metastasis or death and with expected curative outcome (such as adequately treated carcinoma in situ of the cervix, basal or squamous cell skin cancer, or ductal carcinoma in situ treated surgically with curative intent) or undergoing active surveillance per standard-of-care management (e.g., chronic lymphocytic leukemia Rai stage 0) * Any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications * Except for cohort 4, Merkel cell carcinoma, prior treatment with CD137 agonists or immune checkpoint blockade therapies, including anti-CTLA-4, anti-PD-1, and anti-PD-L1 therapeutic antibodies * Treatment with systemic immunostimulatory agents (including, but not limited to, interferon and interleukin 2 \[IL-2\]) within 4 weeks or five half-lives of the drug (whichever is longer) prior to initiation of study treatment * Treatment with systemic immunosuppressive medication (including, but not limited to, corticosteroids, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-TNF-· agents) within 2 weeks prior to initiation of study treatment, or anticipation of need for systemic immunosuppressive medication during the course of the study, with the following exceptions: patients who received low-dose immunosuppressant medication are eligible for the study; patients who received mineralocorticoids (e.g., fludrocortisone), corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenal insufficiency are eligible for the study * Pregnant or breastfeeding, or intending to become pregnant during the study; women of childbearing potential must have a negative serum pregnancy test result within 14 days prior to initiation of study treatment * Inadequately controlled hypertension (defined as systolic blood pressure \> 150 mmHg and/or diastolic blood pressure \> 100 mmHg); anti-hypertensive therapy to maintain a systolic blood pressure \< 150 mmHg and/or diastolic blood pressure \< 100 mmHg is permitted * Prior history of hypertensive crisis or hypertensive encephalopathy * History of stroke or transient ischemic attack within 6 months prior to cycle 1, day 1 * Significant vascular disease (e.g., aortic aneurysm requiring surgical repair or recent peripheral arterial thrombosis) within 6 months prior to cycle 1, day 1 * Patients with a baseline electrocardiography (ECG) demonstrating a corrected QT (QTc) \> 460 ms * Evidence of bleeding diathesis or clinically significant coagulopathy (in the absence of therapeutic anticoagulation) * Current or recent (within 10 calendar days prior to cycle 1, day 1) use of dipyramidole, ticlopidine, clopidogrel, or cilostazol * Core biopsy or other minor surgical procedure, excluding placement of a vascular access device, within 7 calendar days prior to the first dose of bevacizumab * History of abdominal or tracheoesophageal fistula or gastrointestinal perforation within 6 months prior to cycle 1, day 1 * Clinical signs or symptoms of gastrointestinal obstruction or requirement for routine parenteral hydration, parenteral nutrition, or tube feeding * Evidence of abdominal free air not explained by paracentesis or recent surgical procedure * Serious, non-healing or dehiscing wound, active ulcer, or untreated bone fracture * Proteinuria, as demonstrated by urine dipstick or \> 1.0 g of protein in a 24-hour urine collection; all patients with \>= 2+ protein on dipstick urinalysis at baseline must undergo a 24-hour urine collection for protein * Appendiceal adenocarcinoma basket * Complete or partial bowel obstruction * Epstein-Barr virus-associated nasopharyngeal carcinoma basket: * None * Human papilloma virus-associated cancers basket * None * Merkel cell carcinoma basket: * None * Neuroendocrine tumors, pancreatic basket: * Grade 3, poorly differentiated neuroendocrine carcinoma * Large cell or small cell histology * Neuroendocrine tumors, extrapancreatic basket: * Grade 3, poorly differentiated neuroendocrine carcinoma * Large cell or small cell histology * Peritoneal mesothelioma basket: * None * Pleural mesothelioma basket: * None