-  INCLUSION CRITERIA:

               1. Patients must have histopathological confirmation of hepatocellular carcinoma
                  (HCC) or biliary tract carcinoma (BTC) by the Laboratory of Pathology of the NCI
                  prior to entering this study OR histopathological confirmation of carcinoma in
                  the setting of clinical and radiological characteristics which, together with the
                  pathology, are highly suggestive of a diagnosis of HCC (or biliary tract
                  carcinoma). Fibrolamellar variant is also allowed. The term BTC includes intraor
                  extrahepatic cholangiocarcinoma, gallbladder cancer or ampullary cancer.

               2. Patients must have disease that is not amenable to potentially curative
                  resection, transplantation or ablation. HCC patients must have progressed on,
                  been intolerant to, or refused prior sorafenib therapy. Patients with BTC must
                  have received, been intolerant of or refused at least one line of chemotherapy.

               3. Patients must have multiple tumor lesions (at least 2): one for the ablation
                  procedure and another for evaluation located outside the proposal ablation zone.

               4. Disease must be technically amenable to transhepatic arterial chemoembolization
                  (TACE) (HCC patients only), radiofrequency ablation (RFA), or cryoablation. Each
                  case will be discussed at GI tumor board with interventional radiology. Patients
                  must have evaluable disease.

               5. If liver cirrhosis is present, patient must have a Child-Pugh A/B7 classification

               6. Age 18 years or older

               7. ECOG performance status 0-2

               8. Patients must have normal organ and marrow function as defined below:

          -  leukocytes greater than or equal to 3,000/mcL

          -  absolute neutrophil count greater than or equal to 1,000/mcL

          -  platelets greater than or equal to 60,000/mcL

          -  total bilirubin If cirrhosis present: Part of Child Pugh requirement; If no cirrhosis:
             bilirubin should be less than or equal to 2 xULN

          -  serum albumin If cirrhosis present: Part of Child Pugh requirement; If no cirrhosis:
             albumin should be less than or equal to 2.5g/dl

          -  patients are eligible with ALT or AST up to 5 x ULN.

          -  creatinine < 1.5x institution upper limit of normal OR creatinine clearance greater
             than or equal to 45 mL/min/1.73 m^2 for patients with creatinine levels above
             institutional normal

             9. Patients must have recovered from any acute toxicity related to prior therapy,
             including surgery. Toxicity should be less than or equal to grade 1 or returned to
             baseline.

             10. Patients must not have other invasive malignancies within the past 5 years (with
             the exception of non-melanoma skin cancers, non-invasive bladder cancer or localized
             prostate cancer for whom systemic therapy is not required).

             11. Patient must be able to understand and willing to sign a written informed consent
             document.

             12. Evidence of post-menopausal status or negative urinary or serum pregnancy test for
             female pre-menopausal patients. Women will be considered post-menopausal if they have
             been amenorrheic for 12 months without an alternative medical cause. The following
             age-specific requirements apply:

          -  Women <50 years of age would be considered post-menopausal if they have been
             amenorrheic for 12 months or more following cessation of exogenous hormonal treatments
             and if they have luteinizing hormone and follicle-stimulating hormone levels in the
             post-menopausal range for the institution or underwent surgical sterilization
             (bilateral oophorectomy or hysterectomy).

          -  Women greater than or equal to 50 years of age would be considered post-menopausal if
             they have been

        amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments,
        had radiation-induced menopause with last menses >1 year ago, had chemotherapy-induced
        menopause with last menses >1 year ago, or underwent surgical sterilization (bilateral
        oophorectomy, bilateral salpingectomy or hysterectomy. Subject is willing and able to
        comply with the protocol for the duration of the study including undergoing treatment and
        scheduled visits and examinations including follow up.

        13. Body weight greater than 30kg

        EXCLUSION CRITERIA:

          1. Patients who have had standard of care chemotherapy, large field radiotherapy, or
             major surgery must wait 2 weeks prior to entering the study. For recent experimental
             therapies a 28 day period of time must elapse before treatment.

          2. Patients who have undergone prior liver transplantation are ineligible.

          3. Patients with known brain metastases will be excluded from this clinical trial because
             of their poor prognosis and because they often develop progressive neurologic
             dysfunction that would confound the evaluation of neurologic and other adverse events.

          4. Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             systemic infection, symptomatic congestive heart failure, unstable angina pectoris,
             cardiac arrhythmia (excluding insignificant sinus bradycardia and sinus tachycardia)
             or psychiatric illness/social situations that would limit compliance with study
             requirements.

          5. History of chronic autoimmune disease (e.g., Addison s disease, multiple sclerosis,
             Graves disease, Hashimoto s thyroiditis, rheumatoid arthritis, hypophysitis, etc.)
             with symptomatic disease within the 3 years before randomization. Note: Active
             vitiligo or a history of vitiligo will not be a basis for exclusion.

          6. Dementia or significantly altered mental status that would prohibit the understanding
             or rendering of Information and Consent and compliance with the requirements of the
             protocol.

          7. Diverticulitis either active or history of within the past 2 years. Note that
             diverticulosis is permitted.

          8. Active or history of inflammatory bowel disease (colitis, Crohn s), irritable bowel
             disease, celiac disease, or other serious, chronic, gastrointestinal conditions
             associated with diarrhea. Active or history of systemic lupus erythematosus or Wegener
             s granulomatosis. Currently receiving immunosuppressive doses of steroids or other
             immunosuppressive medications (inhaled and topical steroids are permitted)

          9. History of sarcoidosis syndrome

         10. Patients should not be vaccinated with live attenuated vaccines within 30 days of
             starting durvalumab or tremelimumab treatment.

         11. Has a known history of Human Immunodeficiency Virus (HIV). HIV-positive patients
             receiving anti-retroviral therapy are excluded from this study due to the possibility
             of pharmacokinetic interactions between antiretroviral medications and tremelimumab.
             HIV positive patients not receiving antiretroviral therapy are excluded due to the
             possibility that tremelimumab may worsen their condition and the likelihood that the
             underlying condition may obscure the attribution of adverse events.

         12. History of hypersensitivity reaction to human or mouse antibody products.

         13. Pregnancy and breast feeding are exclusion factors. The effects of tremelimumab on the
             developing human fetus are unknown. Enrolled patients must agree to use adequate
             contraception prior to study entry, the duration of study participation and 6 months
             after the end of the treatment. Should a woman become pregnant or suspect she is
             pregnant while she or her partner is participating in this study, she should inform
             her treating physician immediately.

         14. Patients with unhealed surgical wounds for more than 30 days.