Durvalumab and Lenvatinib in Participants With Locally Advanced and Metastatic Hepatocellular Carcinoma ( Dulect2020-1 )

Inclusion Criteria: * Patients must have pathologically or cytologically or by radiological criteria proven hepatocellular carcinoma; known mixed histology (e.g. hepatocellular carcinoma plus cholangiocarcinoma) or fibrolamellar variant is not allowed * Locally advanced and metastatic HCC * Has an eligibility scan (CT of the chest, triphasic CT scan or MRI of the abdomen, and CT or MRI of the pelvis) \<1 week before the treatment of durvalumab in combination with lenvatinib. * Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 7 days prior to Cycle 1, Day 1. * Has a Child-Pugh A liver score (5 to 6 points) within 7 days prior to Cycle1, Day 1. * Has controlled hepatitis B (Hep B) * The estimate time length between enrollment and liver transplantation should be at least 2 months * No prior systemic therapy, local therapy (TACE etc.)\>6w * If female, is not pregnant or breastfeeding, and at least one of the following conditions applies: 1) Is not a woman of childbearing potential (WOCBP); or 2) Is a WOCBP and using a contraceptive method that is highly effective or be abstinent from heterosexual intercourse as their preferred and usual lifestyle (a WOCBP must have a negative pregnancy test within 72 hours before the first dose of study treatment). * Has adequate organ function. * Granulocytes \>= 1,500/uL * Hemoglobin \>= 8.5 g/dL; patients with recent or ongoing gastrointestinal bleed may not be transfused to reach the entry hemoglobin of 8.5 g/dL; physicians should ensure patients requiring transfusion prior to registration do not have an occult or clinically apparent gastrointestinal bleed * Platelets \>= 75,000/uL * Creatinine =\< 1.5 x upper limit of normal (ULN) (or creatinine clearance calculated \>= 60 cc/minute) * Bilirubin =\< 3 mg/dL * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\<5 x ULN * Prothrombin time (PT)-international normalized ratio (INR) =\< 1.7 (not required for patients on anticoagulation agents; patients who are being therapeutically anticoagulated with an agent such as Coumadin or heparin will be allowed to participate provided that no prior evidence of underlying abnormality in these parameters exists) * Patients with a history of hypertension should be well controlled (\< 140/90 mmHg) on a regimen of anti-hypertensive therapy * Significant history of cardiac disease is not allowed: * Congestive heart failure \> class II New York Heart Association (NYHA) Myocardial infarction within 6 months prior to registration Serious myocardial dysfunction, defined as scintigraphically (multigated acquisition scan \[MUGA\], myocardial scintigram) determined absolute left ventricular ejection fraction (LVEF) below 45% or an LVEF on echocardiogram(ECHO) below the normal limit at the individual institution Exclusion Criteria: * Has had esophageal or gastric variceal bleeding within the last 6 months. * Has clinically apparent ascites on physical examination. * Has had clinically diagnosed hepatic encephalopathy in the last 6 months. * Has received local therapy to liver ablation other than with radiofrequency or microwave ablation. * Has a history of (noninfectious) pneumonitis that required steroids or has current pneumonitis * Has an active infection requiring systemic therapy. * Has dual active Hepatitis B Virus (HBV) and Hepatitis C Virus (HCV) infection at study entry. * Has a known history of human immunodeficiency virus (HIV) infection. * Has known active tuberculosis (TB; Bacillus tuberculosis). * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti PD-L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX-40, CD137). * Has received prior systemic anti-cancer therapy for HCC including investigational agents. * Is receiving any of the following prohibited concomitant therapies:1) Antineoplastic systemic chemotherapy or biological therapy; 2) Immunotherapy not specified in this protocol; 3) Investigational agents other than pembrolizumab; 4) Radiation therapy; 5) Oncological surgical therapy; or systemic glucocorticoids for any purpose other than to modulate symptoms from an AE that is suspected to have an immunologic etiology. * Has received a live vaccine within 30 days prior to the first dose of study treatment. * Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to Cycle 1, Day 1. * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to Cycle 1, Day 1. * Has severe hypersensitivity (≥Grade 3) to Durvalumab and/or any of its excipients. * Has an active autoimmune disease that has required systemic treatment in past 2 years. * Has a known psychiatric or substance abuse disorder that would interfere with the participant's ability to cooperate with the requirements of the study. * Is pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of study treatment.