SATURN-STS: Phase II Study of Neoadjuvant Atezolizumab With Doxorubicin, Concurrent Atezolizumab With Pre-operative Radiation Therapy and Adjuvant Atezolizumab in Patients With High-risk Surgically Resectable Extremity and Truncal Soft Tissue Sarcoma

Inclusion Criteria: * Adult subjects (Age ≥18 years) with treatment-naïve de novo or locally recurrent soft tissue sarcomas of the extremity or superficial trunk * Histological diagnosis of grade 2 or grade 3 undifferentiated pleomorphic sarcoma (UPS), myxofibrosarcoma (MFS), de-differentiated/pleomorphic liposarcoma (DDLPS), unclassified sarcoma, and leiomyosarcoma. Pathologic terms for the "unclassified sarcoma" histology may include pleomorphic undifferentiated sarcoma, unclassified spindle cell sarcoma, spindle cell sarcoma not otherwise specified, pleomorphic spindle cell sarcoma, or pleomorphic fibroblastic sarcoma. * Patient must have recent imaging (CT or MRI, as appropriate) within 8 weeks of trial enrollment demonstrating measurable disease, defined as at least one lesion that can be ccurately measured in at least one dimension (longest diameter to be recorded) as \>5 cm. * Patients must have disease determined to be surgically resectable and candidates for upfront surgery as agreed upon by a multidisciplinary consensus (Surgical Oncology, Medical Oncology, Radiation Oncology) after presentation at sarcoma multidisciplinary conference. Resectable tumors are defined as having no significant vascular, neural, or bony involvement. Only cases where a complete surgical resection can safely be achieved are defined as resectable. * Patients must have life expectancy \> 6 months. * ECOG performance status ≤2 (Karnofsky ≥60%) * For patients receiving therapeutic anticoagulation: stable anticoagulant regimen * Patients must have adequate organ and marrow function as defined below: absolute neutrophil count ≥1,000/mcL platelets ≥100,000/mcL total bilirubin ≤ institutional upper limit of normal (ULN) AST(SGOT)/ALT(SGPT) ≤3 × institutional ULN creatinine ≤ eGFR ≥40 ml/min * Negative HIV test at screening, with the following exception: patients with a positive HIV test at screening are eligible provided they are stable on anti-retroviral therapy, have a CD4 count ≥ 200/L, and have an undetectable viral load * Negative hepatitis B surface antigen (HBsAg) test at screening (if relevant) * Negative hepatitis C virus (HCV) antibody test at screening, or positive HCV antibody test followed by a negative HCV RNA test at screening (if relevant) a. The HCV RNA test must be performed for patients who have a positive HCV antibody test. * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better. * The effects of Atezolizumab on the developing human fetus are unknown. For this reason and because doxorubicin as used in this trial is known to be teratogenic, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. (Refer to Pregnancy Assessment Policy MD Anderson Institutional Policy # CLN1114). This includes all female patients, between the onset of menses (as early as 8 years of age) and 55 years unless the patient presents with an applicable exclusionary factor which may be one of the following: * Postmenopausal (no menses in greater than or equal to 12 consecutive months). * History of hysterectomy or bilateral salpingo-oophorectomy. * Ovarian failure (Follicle Stimulating Hormone and Estradiol in menopausal range) * History of bilateral tubal ligation or another surgical sterilization procedure. * For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods, and agreement to refrain from donating eggs, as defined below: * Women must remain abstinent or use contraceptive methods with a failure rate of 1% per year during the treatment period and for 5 months after the final dose of atezolizumab after the final dose of doxorubicin. Women must refrain from donating eggs during this same period. * A woman is considered to be of childbearing potential if she is post-menarchal, 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, fallopian tubes and/or uterus) or another cause as determined by the investigator (e.g., Müllerian agenesis). Per this definition, a woman with a tubal ligation is considered to be of childbearing potential. The definition of childbearing potential may be adapted for alignment with local guidelines or requirements. a. Examples of contraceptive methods with a failure rate of 1% per year include: i. bilateral tubal ligation ii. male sterilization iii. hormonal contraceptives that inhibit ovulation iv. hormone-releasing intrauterine devices v. 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 post ovulation methods) and withdrawal are not adequate methods of contraception. If required per local guidelines or regulations, locally recognized adequate methods of contraception and information about the reliability of abstinence will be described in the local Informed Consent Form. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Disease that is considered surgically unresectable, including, but not limited to significant vascular, neural, or bone involvement, and in cases where a complete surgical resection cannot be safely performed. * Prior chemotherapy or targeted small molecule therapy for the current sarcoma. History of previous systemic treatments for other cancers is allowable. This exclusion criteria also applies to patients who present with locally recurrent disease. * Prior treatment with radiotherapy to the site of the sarcoma (re-irradiation or radiationinduced sarcomas are not allowed on this study) * Metastatic disease or regional lymph node involvement. Chest CT will be mandatory prior to enrollment to evaluate for the presence of metastatic disease. Pulmonary nodule(s) \< 7 mm without a histological diagnosis may not be the basis for studyexclusion giventhelackof specificityof chest CT.If pulmonary nodule(s) measuring 6 - 10 mm are noted on chest CT but appear stable relative to prior chest imaging of at least 6 months duration or if 18FDG-PET scan indicates that the nodule(s) are unlikely to be metastatic disease, then this is permitted. Pulmonary nodules \>10 mm should be considered metastatic unless proven otherwise by biopsy/resection or stable appearance for at least 6 months on imaging. * Active concurrent second malignancy within 2 years of trial enrollment. Note: patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. Examples include non-melanomatous skin cancer, in situ carcinoma, or low-risk prostate cancer. * 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 * Currently receiving cancer therapy (chemotherapy, radiation therapy, immunotherapy, or biologic therapy) or investigational anti-cancer drug within 8 weeks of trial enrollment. * Prior or concurrent immunotherapy, including treatment with an anti-PD-1, anti-PD-L1, antiPD-L2, or anti-CTLA-4 antibody; tumor vaccines; interferon, or interleukins. * Recent stroke or thromboembolic event (within 6 months) requiring anticoagulation that cannot be interrupted. Non- life Threating deep venous thrombosis on stable anticoagulant within 6 months is allowed. * Any serious or uncontrolled medical disorder that, in the opinion of the investigator, may increase the risk associated with study participation or study drug administration, impair the ability of the subject to receive protocol therapy, or interfere with the interpretation of study results. * Subjects with active, known or suspected autoimmune disease including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjögren syndrome, Guillain-Barré syndrome, or multiple sclerosis. a. Patients with a history of autoimmune-related hypothyroidism who are on thyroidreplacement 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: 1. Rash must cover 10% of body surface area 2. Disease is well controlled at baseline and requires only low-potency topical corticosteroids 3. There has been 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 * Significant cardiovascular disease (such as New York Heart Association Class II or greater cardiac disease, myocardial infarction, or cerebrovascular accident) within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina * Current treatment with anti-viral therapy for HBV * Treatment with investigational therapy within 28 days prior to initiation of study treatment * 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 5 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 study treatment, with the following exceptions: 1. Patients who received acute, low-dose systemic immunosuppressant medication or a one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible for the study. 2. 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. * Known hypersensitivity to Chinese hamster ovary cell products or to any component of the atezolizumab formulation * Any positive test result for hepatitis B or C virus indicating acute or chronic infection * Active tuberculosis * History of severe hypersensitivity reaction to any monoclonal antibody * Received a live vaccine within 30 days of planned start of study therapy. Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines, and are not allowed. * Prior allogeneic stem cell or solid organ transplantation * 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 * Unwillingness or inability to follow the procedures required in the protocol * Patients with psychiatric illness/social situations that would limit compliance with study requirements. * Prisoners or subjects who are involuntarily incarcerated * Pregnant women are excluded from this study because doxorubicin a class D agent with the potential for teratogenic or abortifacient effects. * Patients weighing \>400 lbs. as this would disqualify them from undergoing RT (due to physical weight restrictions of the table the patients lie on when being treated with linear accelerators). * 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 study