More than 5 discrete intrahepatic parenchymal foci of definite HCC Measureable common or main branch biliary duct involvement with HCC For HCC: Participants with a first diagnosis of HCC who have undergone a curative resection or ablation Participants are eligible to enroll if they have non-viral related-HCC, or if they have HBV-HCC, or HCV-HCC Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC Participants previously receiving any prior therapy for HCC, including loco-regional therapies HCC for which no other appropriate therapy is available. Note: Expansion Part: No prior systemic therapy for advanced/unresectable HCC Patients must have HCC limited to the liver; there must be no definitive clinical or radiographic evidence of extrahepatic HCC; portal lymphadenopathy is permitted as lymphadenopathy is commonly associated with cirrhosis unrelated to malignancy Patients with locally advanced HCC not eligible for curative therapies Patient who has received previous systemic therapy or transarterial chemoembolization (TACE) for HCC Have a diagnosis of hepatocellular carcinoma (HCC) and high risk for HCC recurrence Concurrent administration of systemic therapy for HCC No prior systemic therapy for HCC Histology and/or cytology confirmed HCC per the enrolling institution; subjects in Cohort 1 are permitted to enroll without confirmation of HCC as long as imaging Liver Imaging Reporting and Data System (LiRAD)s criteria are met and a biopsy is scheduled prior to or the day of the deb-TACE procedure; HCC confirmation must be completed prior to initiation of nivolumab for all cohorts; if a patient is found to not have confirmed HCC, they will be removed from the study Patient must have a histologically confirmed diagnosis hepatocellular carcinoma; known fibrolamellar HCC, sarcomatoid HCC or mixed cholangiocarcinoma and HCC will be excluded Patient who has received previous systemic therapy for HCC Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC. Any prior systemic therapy for HCC Known fibrolamellar or mixed HCC-cholangiocarcinoma histology Confirmed HCC based on histopathological findings from tumor tissues. Unresectable HCC with diagnosis confirmed pathologically or with noninvasive methods. Fibrolamellar carcinoma or mixed HCC Patients must have no prior history of treatment for HCC (treatment naive). Patients must not have received prior anticancer therapy with anti-CLTA-4 or anti-PD1 for HCC. Patients receiving any concomitant systemic therapy for HCC are excluded Prior systemic therapies for HCC are allowed but not required Patients within unresectable HCC Patients with resectable HCC Histological or radiologic diagnosis of advanced (unresectable or metastatic) HCC with Child-Pugh A or Child Pugh B7 cirrhosis:\r\n* The diagnosis of HCC will be made according to the European Association for the Study of the Liver-European Organization for Research and Treatment of Cancer Clinical Practice Guidelines (EASL–EORTC CPG) and according to successive modifications of the American Association for the Study of Liver Disease (AASLD) practice guidelines \r\n* Pathological diagnoses of HCC will be made according to the International Working Party criteria Cytologically or histologically confirmed advanced or metastatic HCC; if no histological diagnosis, patient must have imaging studies compatible with HCC Measurable common or main branch biliary duct involvement with HCC More than one line of prior systemic therapy for HCC No prior treatment of current HCC; however, recurrent HCC after resection may be included Subjects who have radiographic or histological diagnosis of hepatocellular cancer (HCC), with advanced stage disease that is not amenable to curative surgical resection; patients without histologic diagnosis must meet the radiographic criteria for HCC Histological or radiologic confirmation of advanced or metastatic HCC:\r\n* The diagnosis of HCC will be made according to the guidelines of the Barcelona-2000 European Association for the Study of the Liver (EASL) Conference (Bruix et al 2001) and according to successive modifications of the American Association for the Study of Liver Disease (AASLD) (Bruix et al 2005) \r\n* Pathological diagnoses of HCC will be made according to the International Working Party criteria (International Working Party 1995) Diagnosis of HCC: HCC is advanced, ie, treatment-refractory or metastatic, and no standard therapies are expected to be curative. Receipt of no, or of >1, prior systemic drug therapies for HCC. Active malignancy other than HCC. Subjects not meeting the AASLD criteria for HCC will need a biopsy to confirm HCC prior to randomization. Have previously received therapeutic treatment for HCC outside the study protocol or is expected to receive concomitant HCC treatment prior to PFS event. Unilobar HCC Pathologically newly diagnosis HCC, which is deemed resectable and resected No prior systemic therapy for HCC (with the exception of HCC patients enrolled in the safety run-in substudy [Japan only]) Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC histology Patients with advanced HCC not amenable for surgical or loco-regional treatment Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC Subjects who have progressed after or were intolerant to no more than 2 previous systemic therapies for unresectable HCC, or are naïve to systemic therapy. Histological or cytological confirmation of HCC (hepatocellular carcinoma) or non-invasive diagnosis of HCC as per American Association for the Study of Liver Diseases (AASLD) criteria in patients with a confirmed diagnosis of cirrhosis. Diagnosis of liver-only HCC based on European Association for the Study of the Liver (EASL) criteria (radiographic lesion appearance on contrast-enhanced computed tomography [CT] or magnetic resonance imaging [MRI], i.e. enhancement on early arterial phase, washout on portal venous phase with or without associated elevation of serum AFP level > 200 U/ml) or histologic confirmation of HCC diagnosis, whichever is applicable Patients must have histologically diagnosed American Joint Committee on Cancer (AJCC) stage II, III, or IV HCC not eligible for curative resection, transplantation, or ablative therapies\r\n* Cases with mixed, composite, or combined HCC-cholangiocarcinoma histology are eligible with approval from study chair and provided the treating investigator believes it is in the best interest of the patient to treat the tumor with therapy targeted towards the HCC component of tumor based upon review of pathology and clinical\r\ncharacteristics Biliary tract cancers or fibrolamellar variant tumors are excluded\r\n* Cases with mixed, composite, or combined HCC-cholangiocarcinoma histology are eligible with approval from study chair and provided the treating investigator believes it is in the best interest of the patient to treat the tumor with therapy targeted towards the HCC component of tumor based upon review of pathology and clinical characteristics Prior systemic cytotoxic therapies for HCC (chemoembolization is permitted if inclusion criteria are met) Patients must have hepatocellular carcinoma (HCC) with one of the following: \r\n* Microvascular/macrovascular invasion, \r\n* Tumor outside of Milan criteria, \r\n* Poor tumor differentiation; \r\n* Elevated surrogate markers (AFP > 500 or PIVKA [DCP] > 400) pre transplant and with biopsy proven HCC prior to orthotopic liver transplantation (OLT) or on explants Patients with local lymph node metastases\r\n* However, patients with high risk HCC who have been down-staged prior to OLT and show 100% necrosis on explant are eligible as long as there is biopsy proven HCC Patients with fibrolamellar HCC, cholangiocarcinoma, and combined HCC-cholangiocarcinoma Prior use of systemic investigational agents for HCC Phase 2 expansion: HCC Progression following at least 1 prior systemic treatment for HCC. Receipt of more than 2 prior systemic therapies for advanced HCC. Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC Participants with HCC that: Have known HCC with fibro-lamellar or mixed histology. Unresectable HCC confirmed by histology or by non-invasive AASLD criteria Participants with diagnosis of HCC. Subjects must have confirmed diagnosis of unresectable HCC with any of the following criteria: Imaging findings for HCC corresponding to any of the following: HCC with greater than or equal to 50 percent liver occupation Advanced tumoral disease, defined as vascular invasion, extrahepatic spread, or diffuse HCC (50% liver involvement) Subject has a documented diagnosis of advanced HCC of any etiology. Subject has fibrolamellar variant of HCC. Subject has a history of a non-HCC malignancy with the following exceptions: Part A2 only: Patients with histological or cytological diagnosis of HCC who have had 0 to 2 prior lines of systemic therapy (progressed or intolerant to approved HCC standard of care treatment). No prior systemic therapy for HCC Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC Diagnosis of hepatocellular carcinoma (HCC) confirmed histologically, excluding mixed HCC histology (e.g. HCC plus cholangiocarcinoma) or fibrolamellar variant Advanced staged HCC (unresectable and not amenable to local or regional therapy; or metastatic HCC); the diagnosis of HCC should be based on at least one of the following:\r\n* Magnetic resonance imaging (MRI) or computed tomography (CT) consistent with liver cirrhosis AND at least one solid liver lesion measuring >= 2 cm, with characteristics arterial enhancement and venous washout regardless of alpha-fetoprotein (AFP) levels\r\n* AFP >= 400 ng/mL AND evidence of at least one solid liver lesion >= 2 cm regardless of specific imaging characteristics on CT or MRI\r\n* Histological/cytology biopsy confirming HCC Histologically confirmed advanced HCC No prior systemic regimens for HCC Histological or cytological confirmation of HCC (hepatocellular carcinoma) or non-invasive diagnosis of HCC as per American Association for the Study of Liver Diseases criteria in patients with a confirmed diagnosis of cirrhosis Known to be Hepatitis B or C positive (except HCC patients) Have histological evidence of a diagnosis of HCC not amenable to curative surgery Known HCC with fibro-lamellar or mixed histology Diagnosis of HCC Diagnosis of HCC Locally advanced HCC Prior intra-arterial embolization, chemotherapy or systemic therapy for HCC. Early stage hepatocellular carcinoma (HCC) diagnosed based on the typical hallmark of HCC (hypervascular in the arterial phase with washout in the portal venous or delayed phases) Documented complete response to HCC treatment. Diagnosis of fibrolamellar HCC or tumors of mixed histology. Confirmed to have HCC as described by the American Association for the Study of Liver Disease (AASLD). Histologically confirmed HCC Histologically confirmed diagnosis of advanced solid tumor (dose escalation component) or metastatic melanoma (uveal or cutaneous) (doses escalation and MTD expansion components) or platinum-resistant (tumor progression within a year after the completion of platinum-based therapy) ovarian carcinoma (high grade serous, endometrial or poorly differentiated endometrioid) or HCC that has failed treatment with sorafenib or did not tolerate sorafenib or refused sorafenib, or HCC with coexistent BCT that has or has not been treated with chemotherapy, or BCT that has or has not been treated with chemotherapy. For HCC and HCC with coexistent BCT, cirrhotic status of Child-Pugh grade A-B7 must be present. Child-Pugh status should be determined based on clinical findings and laboratory data during the screening period (Appendix C). Subjects on anti-coagulants are to receive 1 point for their INR status, as they are presumed to have a <1.7 baseline PT/INR. Ovarian cancer, or HCC, or HCC with coexistent BCT, or BCT only tissue either from an archived specimen or from a new biopsy of sufficient amount and quality should be available for IHC determination of ASS status to be performed retrospectively for the ovarian cancer, or HCC, or HCC with coexistent BCT, or BCT only cohorts. Subjects with no tissue available would require a biopsy. Fibrolamellar histology HCC, mixed hepatocholangiocarcinoma, hepatic sarcomas and other non-HCC primary liver tumors First line advanced HCC (i.e., no prior systemic therapy) HCC diagnosed by tissue or imaging study Unresectable HCC without extrahepatic disease based on CT No prior systemic therapy for HCC Patients with hepatocellular carcinoma (HCC) are eligible for this trial; HCC is defined as having at least one of the following:\r\n* HCC diagnosed either on biopsy or based on standard imaging criteria on contrast enhanced computed tomography (CT) or magnetic resonance imaging (MRI) (arterial enhancement with washout and pseudocapsule); or\r\n* A discrete hepatic tumor(s) as defined by the Barcelona criteria for cirrhotic patients, > 1 centimeter (cm) with arterial hypervascularity and venous or delayed phase washout on CT or MRI\r\n* Presentation at multidisciplinary liver tumor board to assess eligibility for either SBRT or MWA The foci of HCC must be in an anatomic location amendable to treatment by both MWA and SBRT Patients with 3 or more foci of HCC Participants with advanced or metastatic and/or unresectable HCC A diagnosis of HCC based on histopathologic findings, or a diagnosis of cirrhosis and a tumor with classical HCC imaging characteristics. Participant has confirmed HCC by CT/MRI; participants who have previously had HCC but have been treated and have been recurrence free for 5 years are eligible Have an HCC mass viewable on grayscale B-mode ultrasound Patients not eligible or scheduled for TACE of a HCC mass Patients with a known infiltrative variant of HCC Patients diagnosed with HCC, who will undergo resection or transplantation within 6 months, as part of routine clinical care and patients diagnosed with unresectable HCC HCC PATIENTS: Patient with confirmed diagnosis of HCC, and untreated or HCC PATIENTS: Patients with suspected HCC (suspected HCC nodules should preferably be smaller than 3 cm and preferably within 6 cm in depth of the transducer head to minimize attenuation) and untreated or HCC PATIENTS: Patients at a higher risk of HCC undergoing a screening program by ultrasound Liver tumor diagnosed histologically as HCC or suspected of being HCC in association with serum alpha-fetoprotein level > 200 or tumor mass with characteristics of malignancy on diagnostic imaging Cohort A: HCC with FGF19 amplification. Cohort B: HCC without FGF19 amplification. Histologically confirmed HCC, not amenable to transplant, resection or loco-regional therapy Other active malignancy besides HCC within 3 years. Must be listed for liver transplantation with HCC exception points based on the imaging diagnosis of at least one Organ Procurement and Transplantation Network (OPTN) class 5 HCC lesion(s) per study-defined imaging criteria (participating institutions may not enroll patients in whom the HCC diagnosis is solely based on biopsy and who do not have at least one liver lesion that meets imaging criteria for OPTN Stage 2, Class 5 HCC)\r\n* Patients must meet one of the following descriptions based on imaging findings:\r\n** EITHER OPTN Class 5B: at least 1 focal liver lesion(s) >= 2 cm diameter compatible with imaging diagnosis of Stage II HCC on contrast-enhanced CT imaging and/or contrast-enhanced MRI;\r\n** OR OPTN Class 5A: 2 or 3 focal liver lesions, each between > 1 and < 3 cm diameter, if each is compatible with imaging diagnosis of HCC on contrast-enhanced CT imaging and/or contrast-enhanced MRI\r\n** Imaging findings at the patient level in both situations must be within the United Network for Organ Sharing (UNOS) Stage 2, which is Milan criteria Does not meet OPTN Class 5 imaging criteria for HCC, even if they have biopsy-proven HCC; NOTE: Patients enrolled to the trial under the “Declaration of Intent to List” mechanism who fail to be listed with HCC MELD/PELD Score Exception history data on the UNOS UNET Web site within 60 days from enrollment will come off trial and will not be counted towards target accrual Subjects who have received prior treatment for HCC (prior surgical procedures not related to HCC are allowed) History of HCC or suspicious mass on imaging within 6 months prior to ascertainment of eligibility HCC screening prior to randomization