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

          -  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) with
             conventional techniques or as >= 10 mm (>= 1 cm) with spiral computed tomography (CT)
             scan, magnetic resonance imaging (MRI), or calipers by clinical exam

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

          -  Age >= 6 years at the NCI clinical center (>= 18 years at other participating sites)

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

          -  Life expectancy of greater than 3 months

          -  Leukocytes >= 2,500/mcL

          -  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 >= 30 mL/min/1.73 m^2 by Cockcroft-Gault

          -  Women of child-bearing potential and men 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 study agent; 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

        Exclusion Criteria:

          -  Any prior therapy must have been completed >= 4 weeks or 5 half-lives if known prior
             to enrollment on protocol and the participant must have recovered to eligibility
             levels from prior toxicity; 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
             and all associated toxicities resolved to eligibility levels; patients who have had
             prior monoclonal antibody therapy must have completed that therapy at least 3
             half-lives of the antibody or 6 weeks ago; patients who have received more than a
             cumulative dose of 350 mg/m^2 of doxorubicin may be enrolled at the discretion of the
             coordinating center principal investigator (PI), with a screening echocardiogram

          -  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,
                  provided the following requirements 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)

          -  Treatment with any other investigational agent within 4 weeks prior to cycle 1, day 1;
             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 subtherapeutic dose of drug is administered) at the coordinating center
             principal investigator's (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) 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 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, cerebellum, or within
                       10 mm of the optic apparatus (optic nerves and chiasm)

                    -  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 the screening radiographic 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 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; 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 > 500 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 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

          -  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

          -  Patients requiring treatment with a RANKL inhibitor (e.g. denosumab) who cannot
             discontinue it before treatment with atezolizumab