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 Maximal diameter of any one hepatocellular carcinoma > 15 cm Upper age limit of =< 18 years of age for medullary thyroid carcinoma (MTC), renal cell carcinoma (RCC) and hepatocellular carcinoma (HCC) Patients must have confirmed hepatocellular carcinoma (HCC) by histopathology or imaging criteria according to AASLD guidelines. Confirmed diagnosis of hepatocellular carcinoma (HCC) Subjects must have a diagnosis of measurable advanced or metastatic hepatocellular carcinoma; advanced HCC is defined as disease not amenable to surgery, ablation, transplant, or embolic therapy Histologically confirmed diagnosis of hepatocellular carcinoma (mixed HCC/cholangiocarcinoma is allowed) Hepatocellular carcinoma; HCC (unresectable hepatocellular carcinoma) histopathological diagnosis confirmation based on tumor tissue Confirmed hepatocellular carcinoma (HCC) by one of the following: \r\n* Histopathology\r\n* One radiographic technique that confirms a lesion >= 1 cm with arterial hypervascularization with washout on delayed phase COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Patients with hepatocellular carcinoma do not need to have histologic confirmation of disease as long as they meet the radiologic criteria for diagnosis of hepatocellular carcinoma (HCC) (evidence of arterial phase enhancement with corresponding venous or delayed phase wash out) COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Patients must have advanced disease that is not amenable for resection or transplantation, and that is not treatable with liver directed modalities such as radiofrequency ablation or transarterial chemoembolization COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Patients are not required to have failed sorafenib COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Ascites that is not medically controlled or that required a therapeutic paracentesis within last 3 months COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Any episode of hepatic encephalopathy within the previous 6 months COHORT B, GROUP 6: HEPATOCELLULAR CARCINOMA: Variceal bleeding within last 6 months Histologically or cytologically confirmed hepatocellular carcinoma or biliary tract cancer. Progressed on, be intolerant of, or refused sorafenib (for hepatocellular carcinoma [HCC]), second line treatment and beyond for cholangiocarcinoma or gemcitabine-based chemotherapy for biliary tract cancer. Histologic or cytologic diagnosis of unresectable, locally advanced and/or metastatic hepatocellular carcinoma (HCC) not amenable to curative surgery, transplantation, or ablative therapies based upon assessment of treating investigator Confirmed unresectable or metastatic hepatocellular carcinoma; confirmation either by histologic confirmation or accepted radiographic criteria Diagnosis of primary liver malignancy including hepatocellular carcinoma (HCC) or cholangiocarcinoma by characteristic imaging findings on computed tomography (CT) or MRI, clinical presentation, and/or pathologic confirmation of diagnosis; subjects with other current or prior malignancies are eligible for this study Diagnosis of hepatocellular carcinoma Pathologic diagnosis of hepatocellular carcinoma (including fibrolamellar variants and biphenotypic tumors with a hepatocellular carcinoma [HCC] component) Patients with diagnoses of hepatocellular carcinoma (HCC) according to European Association for the Study of Liver disease (EASL) criteria for diagnosis; regional lymphadenopathy will be allowed\r\n* Any virus status accepted (e.g. hepatitis C, etc.)\r\n* Any prior liver treatment Colorectal, hepatocellular, gallbladder, cholangiocarcinoma, neuroendocrine, melanomas, hematological and central nervous system (CNS) malignancies; Presence of primary hepatobiliary malignancy, including cholangiocarcinoma or hepatocellular carcinoma, gallbladder carcinoma, cancer of ampulla of Vater. Advanced, unresectable hepatocellular carcinoma (unsuitable for resection, transplant or ablation) Any one hepatocellular carcinoma > 15 cm Total maximal sum of hepatocellular carcinomas or a single conglomerate HCC > 20 cm Subjects with advanced hepatocellular carcinoma (HCC) with no curative option Patient must have advanced hepatocellular carcinoma; fibrolamellar HCC is not allowed; hepatocellular carcinoma should be confirmed by at least one of the following:\r\n* Tissue diagnosis\r\n* The presence of one or more liver lesions measuring >= 2 cm in longest diameter, showing characteristic arterial enhancement and venous washout using arterial-phase contrast enhanced imaging, and a clinical history of cirrhosis Diagnosis of hepatocellular carcinoma, cholangiocarcinoma, or liver metastasis from any histology Part 2 Group B is restricted to HCC (fibrolamellar and mixed hepatocellular/cholangiocarcinoma subtypes are not eligible). Patients must have a diagnosis of hepatocellular carcinoma by at least one criterion listed below:\r\n* Histologically confirmed\r\n* Magnetic resonance imaging (MRI) or computerized tomography (CT) findings consistent with hepatocellular carcinoma\r\n* Alpha fetoprotein (AFP) > 400 ng/mL AND evidence of at least one solid liver lesion > 2 cm regardless of specific imaging characteristics on CT or MRI Histologically or cytologically confirmed hepatocellular carcinoma (HCC) or clinical diagnosis by American Association for the Study of Liver Diseases (AASLD) criteria in cirrhotic subjects is required Patients must have a diagnosis of hepatocellular carcinoma confirmed by at least one of the following: \r\n* Histological confirmation; \r\n* Imaging results consistent with cirrhosis and at least one solid liver lesion of > 2 cm with early enhancement and delayed washout (American Association for the Study of Liver Diseases [AASLD] criteria for diagnosis of hepatocellular carcinoma [HCC]);\r\n* Alpha fetoprotein level > 400 ng/mL and evidence of at least one solid liver lesion > 2 cm, regardless of specific imaging characteristics on magnetic resonance imaging (MRI) Patient diagnosed with hepatocellular carcinoma in both lobes of the liver by one of the following methods\r\n* Pathologically confirmed hepatocellular carcinoma (HCC) by biopsy, OR\r\n* HCC > 2 cm with classic radiographic findings of arterial phase enhancement with venous phase washout and pseudocapsule formation on contrast enhanced magnetic resonance imaging (MRI) or computed tomography (CT), OR\r\n* Lesion greater than 2 cm with probable imaging features of HCC and imaging findings of cirrhosis and/or portal hypertension or a serum alpha fetoprotein (AFP) greater than 200 ng/mL Patient diagnosed with hepatocellular carcinoma and listed for orthotopic liver transplantation at the participating institution in accordance with Organ Procurement and Transplantation Network (OPTN) guidelines Patients with hepatocellular carcinoma will be excluded from this study Subjects with hepatocellular carcinoma must have received sorafenib as one of the standard treatment options prior to being enrolled into the study Subject has a confirmed pathologic diagnosis of Hepatocellular carcinoma (HCC) according to the American Association for the Study of Liver Diseases (AASLD) Guidelines. Subject has received more than 2 previous systemic therapies for Hepatocellular carcinoma (HCC). Pathologically (histologically or cytologically) or radiographically-proven (based on the American Association for the Study of Liver Diseases [AALSD] criteria) unresectable or locally recurrent hepatocellular cancer prior to registration Patients with hepatocellular carcinoma are eligible for this trial; hepatocellular carcinoma is defined as having at least one of the following:\r\n* Biopsy proven hepatocellular carcinoma (HCC); or\r\n* A discrete hepatic tumor(s) as defined by the Barcelona criteria – for cirrhotic patients, > 1 cm with arterial hypervascularity and venous or delayed phase washout on computed tomography (CT) or MRI If the patient is determined to be cirrhotic (based on criteria outlined earlier), the patient must have an ultrasound done within 6 months prior to enrollment with no evidence of hepatocellular carcinoma Has a HCC diagnosis confirmed by radiology, histology or cytology (fibrolamellar and mixed hepatocellular/cholangiocarcinoma subtypes are not eligible). Must have histologically or cytologically confirmed diagnosis of HCC (fibrolamellar and mixed hepatocellular/cholangiocarcinoma subtypes are not eligible). Chronic hepatitis (except for subjects with hepatocellular carcinoma) Fibrolamellar carcinoma or mixed hepatocellular cholangiocarcinoma. Hepatocellular carcinoma (HCC)\r\n* Not eligible for curative attempt resection or liver transplant Histologically confirmed advanced hepatocellular carcinoma, not eligible for surgical and/or locoregional therapies; or progressive disease after surgical and /or locoregional therapies Locoregional therapy for hepatocellular carcinoma (HCC) must be completed at least 4 weeks prior to the baseline scan Cohort 2: Participants must have hepatocellular carcinoma (HCC). These participants should have Child-Pugh stage A. Subjects of 18 years or older (men and women) with histologically confirmed advanced hepatocellular carcinoma, not eligible for surgical and/or locoregional therapies; or progressive disease after surgical and /or locoregional therapies Patients with hepatocellular carcinoma may be eligible provided they have AST and ALT that are ? 5.0 x ULN. Platelets ? 100 x 10^9/L; patients with hepatocellular carcinoma may enroll provided they have a platelet count ? 75 x 10^9/L. Have mixed hepatocellular biliary tract cancer histology. For Cohort 1: has histologically or cytologically confirmed diagnosis of HCC (fibrolamellar and mixed hepatocellular/cholangiocarcinoma subtypes are not eligible) based on pathology report For Cohort 2: has an HCC diagnosis confirmed by radiology, histology, or cytology (fibrolamellar and mixed hepatocellular/cholangiocarcinoma subtypes are not eligible) Platelets ? 100 x 10^9/L; patients with hepatocellular carcinoma may enroll provided they have a platelet count ? 75 x 10^9/L. Patients with the diagnosis of hepatocellular carcinoma (HCC) currently being evaluated for liver transplantation and considered for downstaging History of recurrent bacterial infections unrelated to hepatocellular carcinoma (HCC) (particularly skin or lung) Hepatocellular carcinoma for which treatment with FOLFOX6 or CAPOX would be acceptable as determined by the Investigator. Locally advanced or metastatic and/or unresectable Hepatocellular Carcinoma (HCC) Histologically/cytologically verified, non-resectable, recurrent, or metastatic biliary tract carcinoma including intrahepatic cholangiocarcinoma, extrahepatic cholangiocarcinoma and gallbladder carcinoma; combined cholangiocarcinoma and hepatocellular carcinoma is allowed Subject has a confirmed pathologic diagnosis of Hepatocellular carcinoma according to the American Association for the Study of Liver Diseases Guidelines.A biopsy performed at screening may serve as a diagnostic biopsy for subjects with radiographic diagnosis. Histologic diagnosis of liver-confined fibrolamellar or non-fibrolamellar hepatocellular carcinoma (HCC) Advanced unresectable hepatocellular carcinoma with branch portal vein thrombosis (confirmed by non-invasive criteria EASL/AASLD, mandatory by histology in non-cirrhotic patients); can be naive or recurrent HCC after curative treatment (> 6 months before randomization) Patients (except those with hepatocellular carcinoma) must have histologically or cytologically confirmed (at original diagnosis or subsequent recurrence or progression) solid tumor or lymphoma that is metastatic, unresectable, progressive, or recurrent, and for which standard curative or palliative measures do not exist or are no longer effective; patients with hepatocellular carcinoma do not require biopsy confirmation; a liver mass with raised alpha-fetoprotein level (>= 500 ng/mL), consistent radiographic changes, and serology and viral deoxyribonucleic acid (DNA)/ribonucleic acid (RNA) measurements consistent with chronic hepatitis will be sufficient to identify hepatocellular carcinoma without the need for pathologic confirmation of the diagnosis; patients with hepatocellular carcinoma must still, however, have disease that has failed standard therapy; having chronic hepatitis B or C will not exclude patients from participating Part B2: Hepatocellular cancer (excluding fibrolamellar carcinoma) Subjects must have histologically or cytologically confirmed advanced solid tumor. However, Hepatocellular Carcinoma (HCC) subjects are allowed without histological confirmation as long as there is radiological diagnosis as per standard criteria Histologically documented hepatocellular carcinoma Histologically or cytologically confirmed hepatocellular carcinoma that is metastatic, unresectable, or recurrent. Patients who have a diagnosis of hepatocellular carcinoma made through radiologic imaging may be eligible, provided they meet the criteria according to the American Association for the Study of Liver Disease, AASLD (Bruix and Sherman, 2005; Bruix and Sherman, 2011) Cytological or histological confirmed diagnosis of advanced hepatocellular or renal cell carcinoma; HCC patients should not be amenable to treatment with surgery or to orthotopic liver transplant (Phase I) Subjects with hepatocellular carcinoma Hepatocellular Carcinoma (HCC) --Histologic or cytologic diagnosis of hepatocellular carcinoma Patients diagnosed with hepatocellular carcinoma, or who have a history of biliary sepsis within the past 2 years Barcelona Clinic Liver Cancer (BCLC) advanced stage (C) hepatocellular carcinoma, or BCLC intermediate stage (B) hepatocellular carcinoma if treatment with transarterial chemoembolization is not considered appropriate PATIENT: Biopsy and/or radiograph proven diagnosis of hepatocellular carcinoma, cholangiocarcinoma, gallbladder carcinoma or breast, ovarian, or colorectal cancer with liver metastases with a life expectancy of at least one year Fibrolamellar carcinoma or mixed hepatocellular cholangiocarcinoma Fibrolamellar carcinoma or mixed hepatocellular cholangiocarcinoma. Patients will have one of the following diseases: primary hepatocellular cancer, hepatobiliary cancer, or metastatic disease to the liver Prior known or suspected hepatocellular carcinoma Only participants found to express high levels (immunohistochemistry [IHC] score 3 and above) of gamma-OHPdG (gamma-OHPdG-high hepatocellular carcinoma [HCC]) in baseline liver biopsy will be enrolled to receive Polyphenon E treatment Hepatocellular carcinoma Patients with a diagnosis of hepatocellular carcinoma (HCC), intrahepatic cholangiocarcinoma, or metastatic liver cancer planned to receive definitive doses of radiation or surgical resection are eligible Scheduled for TACE treatment of a hepatocellular carcinoma (HCC) mass (lesions reported as Liver Imaging Reporting and Data Systems 4B or 5 or Organ Procurement and Transplantation Network 5a or 5b) Subjects must have a diagnosis of hepatocellular carcinoma (HCC) and a treatment plan to undergo radioembolization therapy with Y-90 at Indiana University Health Hospital Histopathologic or imaging and clinical features of tumor(s) diagnostic for hepatocellular carcinoma with at least one tumor >= 1.5 cm; imaging features diagnostic for hepatocellular carcinoma will be defined as Liver Imaging Reporting and Data System (LI-RADS) 4 or greater Locoregional treatment of hepatocellular carcinoma within the prior 3 months or chemotherapy within the previous 3 months Subjects must have verified unresectable hepatocellular carcinoma (HCC), diagnosed on the basis of clinical and imaging criteria Participants must have histologically or cytologically confirmed hepatocellular carcinoma OR have an imaging study that demonstrates a focal hepatic lesion with imaging features diagnostic of hepatocellular carcinoma