Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed alveolar soft part
             sarcoma that is not curable by surgery. Diagnosis of malignancy must be confirmed by
             the department of pathology at the institution where the patient is enrolled prior to
             patient enrollment. To be eligible for atezolizumab in combination with bevacizumab,
             the patient must have atezolizumab-refractory/resistant disease that has progressed
             (definitive clinical progression or iCPD) on prior 10005 atezolizumab monotherapy

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as >= 20 mm (>= 2 cm) by chest
             x-ray or as >= 10 mm (>= 1 cm) with computed tomography (CT) scan, magnetic resonance
             imaging (MRI), or calipers by clinical exam

          -  Patients with newly diagnosed, unresectable, metastatic and measurable ASPS will also
             be eligible for atezolizumab monotherapy if they show clinical evidence of disease
             progression (including history and increasing physical symptoms); on-study
             documentation will include a physician's rationale that supports evidence of clinical
             disease progression (i.e., increasing tumor pain)

          -  To be eligible for atezolizumab monotherapy, subjects must be >= 2 years of age at the
             National Cancer Institute (NCI) clinical center (>= 14 years of age at other
             participating sites). To be eligible for atezolizumab in combination with bevacizumab,
             subjects must be >=18 years of age

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky or Lansky
             >= 70%)

          -  Life expectancy of greater than 3 months

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Hemoglobin >= 8 g/dL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (however, patients
             with known Gilbert disease who have serum bilirubin level =< 3 x ULN may be enrolled)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase [SGPT]) =<
             3 x ULN (AST and/or ALT =< 5 x ULN for patients with liver involvement)

          -  Alkaline phosphatase =< 2.5 x ULN (=< 5 x ULN for patients with documented liver
             involvement or bone metastases)

          -  Creatinine clearance for adult patients (>= 18 years of age): >= 30 mL/min/1.73 m^2 by
             Cockcroft-Gault; for pediatric patients (< 18 years of age), a serum creatinine based
             on age and gender as follows:

               -  Age 2 to < 6 years, maximum serum creatinine (mg/dL) male 0.8, female 0.8;

               -  Age 6 to < 10 years, maximum serum creatinine (mg/dL) male 1, female 1;

               -  Age 10 to < 13 years, maximum serum creatinine (mg/dL) male 1.2, female 1.2;

               -  Age 13 to < 16 years, maximum serum creatinine (mg/dL) male 1.5, female 1.4;

               -  Age 16 to < 18 years, maximum serum creatinine (mg/dL) male 1.7, female 1.4

          -  Administration of atezolizumab or bevacizumab may have an adverse effect on pregnancy
             and poses a risk to the human fetus, including embryo-lethality. Female patients of
             child-bearing potential and male patients must agree to use adequate contraception
             (hormonal or barrier method of birth control; abstinence) prior to study entry, for
             the duration of study participation, and for 5 months (150 days) after the last dose
             of atezolizumab or 6 months (180 days) after the last dose of bevacizumab. Should a
             woman become pregnant or suspect she is pregnant while she or her partner is
             participating in this study, she should inform her treating physician immediately

          -  Ability to understand and the willingness to sign a written informed consent document
             or a parent/guardian able to do the same

          -  Willingness to provide biopsy samples for research purposes (patients >= 18 years of
             age only)

        Exclusion Criteria:

          -  Any prior therapy must have been completed >= 4 weeks or, if known, >= 5 half-lives of
             the prior agent (whichever is shorter) prior to enrollment on protocol (minimum of 1
             week between prior therapy and study enrollment), and the participant must have
             recovered to eligibility levels from prior toxicity. Note: For patients being treated
             on the atezolizumab plus bevacizumab arm, there is no washout requirement for prior
             anti-PD-1 or anti-PD-L1 agents. Patients should be at least 6 weeks out from
             nitrosoureas and mitomycin C. Prior definitive radiation should have been completed >=
             4 weeks or palliative radiation should have been completed >= 2 weeks prior to study
             enrollment or crossover and all associated toxicities resolved to eligibility levels
             (patients on study may be eligible for palliative radiotherapy to non-targeted lesions
             after 2 cycles of therapy at the principal investigator [PI]'s discretion). Patients
             who have had prior monoclonal antibody therapy must have completed that therapy >= 6
             weeks (or 3 half-lives of the antibody, whichever is shorter) prior to enrollment on
             protocol (minimum of 1 week between prior therapy and study enrollment). A patient who
             has received a cumulative dose of > 350 mg/m^2 of anthracycline (regardless of
             cardioprotectant) may only be enrolled if their ejection fraction measured by an
             echocardiogram is within normal institutional limits

          -  Prior treatment with anti-PD-1 or anti-PD-L1 therapeutic antibody, or
             pathway-targeting agents

               -  Patients who have received prior treatment with anti-CTLA-4 may be enrolled on
                  the atezolizumab monotherapy arm, provided the following requirements and all
                  other selection criteria are met:

                    -  Minimum of 12 weeks from the first dose of anti-CTLA-4 and > 6 weeks from
                       the last dose

                    -  No history of severe immune-related adverse effects from anti-CTLA-4 (NCI
                       Common Terminology Criteria for Adverse Events [CTCAE] grade 3 and 4)

               -  Patients who have progressed on P10005 atezolizumab monotherapy may be eligible
                  for the atezolizumab plus bevacizumab arm, provided the following requirements,
                  and all other selection criteria, are met:

                    -  No prior disease progression on an immune checkpoint inhibitor plus tyrosine
                       kinase inhibitor combination therapy

                    -  Any prior toxicities have resolved to eligibility levels

                    -  No contraindications

          -  Treatment with any other investigational agent within 4 weeks (or within five
             half-lives of the investigational product, whichever is shorter) prior to cycle 1, day
             1 (minimum of 1 week between prior therapy and study enrollment); patients must be >=
             2 weeks since any investigational agent administered as part of a phase 0 study (also
             referred to as an "early phase I study" or "pre-phase I study" where a sub-therapeutic
             dose of drug is administered) at the coordinating center PI's discretion, and should
             have recovered to eligibility levels from any toxicities

          -  Treatment with systemic immunostimulatory agents (including, but not limited to,
             interferon-alpha or interleukin-2 [aldesleukin]) within 6 weeks prior to cycle 1, day
             1.

          -  Treatment with systemic immunosuppressive medications (including, but not limited to,
             prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor
             necrosis factor [anti-TNF] agents) within 2 weeks prior to cycle 1, day 1

               -  Patients who have received acute, low dose, systemic immunosuppressant
                  medications (e.g., a one-time dose of dexamethasone for nausea) may be enrolled

               -  The use of inhaled corticosteroids and systemic mineralocorticoids (e.g.,
                  fludrocortisone) for patients with orthostatic hypotension or adrenocortical
                  insufficiency is allowed

          -  Patients taking bisphosphonate therapy for symptomatic hypercalcemia; use of
             bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is
             allowed

          -  Patients with known primary central nervous system (CNS) malignancy or symptomatic CNS
             metastases are excluded, with the following exceptions:

               -  Patients with asymptomatic untreated CNS disease may be enrolled, provided all of
                  the following criteria are met:

                    -  Evaluable or measurable disease outside the CNS

                    -  No metastases to brain stem, midbrain, pons, medulla, or cerebellum

                    -  No history of intracranial hemorrhage or spinal cord hemorrhage

                    -  No ongoing requirement for dexamethasone for CNS disease; patients on a
                       stable dose of anticonvulsants are permitted

                    -  No neurosurgical resection or brain biopsy within 28 days prior to cycle 1,
                       day 1

               -  Patients with asymptomatic treated CNS metastases may be enrolled, provided all
                  the criteria listed above are met as well as the following:

                    -  Radiographic demonstration of improvement upon the completion of
                       CNS-directed therapy and no evidence of interim progression between the
                       completion of CNS-directed therapy and radiographic screening for the
                       current study

                    -  No stereotactic radiation or whole-brain radiation within 28 days prior to
                       cycle 1, day 1

                    -  Screening CNS radiographic study >= 4 weeks from completion of radiotherapy
                       and >= 2 weeks from discontinuation of corticosteroids

          -  Known hypersensitivity to Chinese hamster ovary cell products or other recombinant
             human antibodies

          -  History of severe allergic, anaphylactic, or other hypersensitivity reactions to
             chimeric or humanized antibodies (i.e., antibodies with generic names ending in
             "ximab" or "zumab", respectively) or fusion proteins

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because atezolizumab is an investigational
             agent with the unknown potential for teratogenic or abortifacient effects; because
             there is an unknown but potential risk for adverse events in nursing infants secondary
             to treatment of the mother with atezolizumab, breastfeeding should be discontinued if
             the mother is treated with atezolizumab

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible; patients must also have a CD4 count > 350 cells/mm^3 with an
             undetectable viral load

          -  Patients on supraphysiologic doses of steroids or use of such within the previous six
             weeks

          -  Known clinically significant liver disease, including active viral, alcoholic, or
             other hepatitis; cirrhosis; fatty liver; and inherited liver disease.

               -  Patients with past or resolved hepatitis B infection (defined as having a
                  negative hepatitis B surface antigen [HBsAg] test and a positive anti-HBc
                  [antibody to hepatitis B core antigen] antibody test) are eligible

               -  Patients positive for hepatitis C virus (HCV) antibody are eligible only if
                  polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA)

          -  History or risk of autoimmune disease, including, but 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

               -  Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid
                  replacement hormone may be eligible

               -  Patients with controlled type 1 diabetes mellitus on a stable insulin regimen may
                  be eligible

               -  Patients with eczema, psoriasis, lichen simplex chronicus of vitiligo with
                  dermatologic manifestations only (e.g., patients with psoriatic arthritis would
                  be excluded) are permitted provided that they meet the following conditions:

                    -  Patients with psoriasis must have a baseline ophthalmologic exam to rule out
                       ocular manifestations

                    -  Rash must cover less than 10% of body surface area (BSA)

                    -  Disease is well controlled at baseline and only requiring low potency
                       topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%,
                       fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%)

                    -  No acute exacerbations of underlying condition within the last 12 months
                       (not requiring psoralen plus ultraviolet A radiation [PUVA], methotrexate,
                       retinoids, biologic agents, oral calcineurin inhibitors; high potency or
                       oral steroids)

          -  History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced),
             organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing
             pneumonia, etc.), or evidence of active pneumonitis on screening chest computed
             tomography (CT) scan; history of radiation pneumonitis in the radiation field
             (fibrosis) is permitted

          -  Patients with active tuberculosis (TB) are excluded

          -  Severe infections within 4 weeks prior to cycle 1, day 1, including, but not limited
             to, hospitalization for complications of infection, bacteremia, or severe pneumonia

          -  Signs or symptoms of infection within 2 weeks prior to cycle 1, day 1

          -  Received oral or intravenous (IV) antibiotics within 2 weeks prior to cycle 1, day 1;
             patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract
             infection or chronic obstructive pulmonary disease) are eligible

          -  Major surgical procedure within 28 days prior to cycle 1, day 1 or anticipation of
             need for a major surgical procedure during the course of the study. Note: Patients
             with a surgical procedure (excluding biopsy procedures) within 28 days prior to cycle
             1 day 1, or any unhealed surgical wound, are not eligible for the atezolizumab +
             bevacizumab combination arm

          -  Administration of a live, attenuated vaccine within 4 weeks before cycle 1, day 1 or
             anticipation that such a live, attenuated vaccine will be required during the study
             and up to 5 months after the last dose of atezolizumab

               -  Influenza vaccination should be given during influenza season only (approximately
                  October to March); patients must not receive live, attenuated influenza vaccine
                  within 4 weeks prior to cycle 1, day 1 or at any time during the study

          -  BEVACIZUMAB-SPECIFIC EXCLUSION CRITERIA

          -  Age < 18 years

          -  Prior disease progression on an immune checkpoint inhibitor plus tyrosine kinase
             inhibitor combination therapy

          -  Major surgery within 6 weeks of enrollment. Minor procedures (e.g. port placement,
             endoscopy with intervention) within 28 days of enrollment (excluding biopsy
             procedures)

          -  Chronic daily treatment with a non-steroidal anti-inflammatory drug (NSAID),
             clopidogrel, dypiridamole, or aspirin therapy > 81 mg/day. Treatment with any such
             agents within 7 days prior to cycle 1 day 1

          -  2+ protein on urinalysis, followed by 24-hour protein of > 1 g

          -  Patients with a history of bleeding varices within 1 year of enrollment

          -  Thromboembolic event within 6 months of enrollment (including cerebrovascular accident
             [CVA] and myocardial infarction (MI)

          -  Evidence of bleeding diathesis or significant coagulopathy (with or without current
             therapeutic anticoagulation)

          -  International normal