Immunotherapy for Advanced Liver Cancer

Inclusion Criteria: * PRIOR TO STEP 1 REGISTRATION: * Diagnosis of hepatocellular carcinoma (HCC) by at least 1 criterion listed below: * Pathologically (histologically or cytologically) proven diagnosis of HCC (strongly recommended) * Radiographically proven (American Association for the Study of Liver Diseases \[AASLD\] criteria) diagnosis of HCC by multiphasic MRI and/or CT scan is allowed. * For patients with a prior or concurrent malignancy, pathologic confirmation of hepatocellular cancer is required. * HCC macrovascular invasion, defined as enhancing vascular thrombosis demonstrating arterial enhancement and venous or delayed venous washout on multiphasic MRI and/or CT is required. * Presence of extrahepatic metastatic disease on CT chest and CT or MRI pelvis, or PET/CT chest/abdomen/pelvis is permitted. * 5 or fewer discrete intrahepatic parenchymal foci of HCC. * Total maximal sum of hepatocellular carcinoma tumors, as a single conglomerate, multiple lesions, or infiltrative HCC \< 20 cm in total summed diameter. * No direct primary tumor extension into the stomach, duodenum, small bowel, or large bowel. * No known fibrolamellar HCC, sarcomatoid HCC, or biphenotypic HCC. * Child-Pugh class A or B7 liver function. * Age ≥ 18. * Eastern Cooperative Oncology Group (ECOG) performance status of 0-2. * Not pregnant and not nursing * Negative urine or serum pregnancy test (in persons of childbearing potential) within 30 days prior to registration. Childbearing potential is defined as any person who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm\^3. * Platelets ≥ 60,000 cells/mm\^3. * Hemoglobin ≥ 8g/dl (Note: The use of transfusion or other intervention to achieve hemoglobin \[Hgb\] ≥ 8g/dl is acceptable). * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 6 x institutional upper limit of normal (ULN). * Total bilirubin \< 4 x institutional ULN. * Creatinine clearance (CrCL) ≥ 30 mL/min/1.73 m\^2 by the Cockcroft-Gault formula. * For treatment of HCC: * Prior surgical resection, transarterial chemoembolization (TACE), and ablation are permitted. * No prior systemic therapy or transarterial radioembolization (TARE) for HCC. * No history of liver transplantation. * For prior treatment for any malignancy: * Prior systemic therapy for a different cancer is allowable, except for prior immunotherapy. * No prior radiotherapy to the region of the study cancer that would result in significant overlap of radiation therapy fields that would lead to excessive cumulative toxicity at the discretion of the investigator. * No medical contraindication to the standard of care immunotherapy. * For patients to be treated with atezolizumab/bevacizumab: * No history of a gastrointestinal (GI) bleed or other clinically significant bleeding event within 6 months prior to study registration. * Systemic immunostimulatory agents (including, but not limited to, interferons and interleukin-2 \[IL-2\]) are prohibited within 4 weeks or five drug elimination half-lives (whichever is longer) prior to registration and during the study period. * No history of allergic reaction to the systemic therapy agent(s), compounds of similar chemical or biologic composition to the systemic therapy agent(s) (or any of its excipients). * PRIOR TO STEP 2 RANDOMIZATION: * Obtain confirmation of payment coverage (insurance or other) for both possible treatment arms.