Inclusion Criteria:

          -  Histologically or cytologically confirmed hepatocellular carcinoma or biliary tract
             cancer

          -  Locally advanced/unresectable or metastatic disease

          -  Age ≥ 18 years at time of study entry

          -  ECOG Performance Status ≤ 1

          -  One previously unirradiated lesion amenable to 8 Gy x 3 radiotherapy based on
             dosimetric organ tolerance AND another unirradiated measurable lesion (per irRECIST)
             outside of the radiation field

          -  Immunotherapy-naïve

          -  Progressed on, be intolerant of, or refused sorafenib [for HCC], second line treatment
             and beyond for cholangiocarcinoma or gemcitabine-based chemotherapy for biliary tract
             cancer

          -  The benefits of sorafenib have been discussed with the patient and the patient has
             refused treatment with sorafenib.

          -  Viral status (Hepatitis B and C) must be known. All HBV-positive patients must be on
             antiviral medication for viral suppression.

               -  Patients with concomitant HBV infection must have a confirmed diagnosis of HBV
                  characterized by the presence of hepatitis B core antibodies, and be sufficiently
                  suppressed with active antiviral treatment (per local institutional practice)
                  prior to enrollment to ensure adequate viral suppression (HBV deoxyribonucleic
                  acid [DNA] <2000 IU/mL).

               -  Patients with concomitant HCV infection must have confirmed diagnosis of HCV
                  characterized by the presence of detectable HCV ribonucleic acid (RNA or anti-HCV
                  antibody upon enrollment.

          -  Body weight ≥ 30 kg

          -  Child-Pugh Score of A. A score of B7 is allowed without severe ascites or without
             hepatic encephalopathy.

          -  Adequate organ and marrow function, defined as:

               -  Hemoglobin ≥ 9 g/dL

               -  ANC ≥ 1.5 x 10^9/L

               -  Platelet count ≥75×109/L

               -  Serum bilirubin ≤2.0× the upper limit of normal (ULN).

               -  ALT and AST ≤ 3 x institutional ULN

               -  Albumin > 2.8 g/dL

               -  INR < 2.0

               -  Calculated creatinine clearance > 40 mL/min as determined by Cockcroft-Gault
                  (using actual body weight)

          -  Males:

             --Creatinine CL (mL/min) = Weight (kg) x (140 - Age) 72 x serum creatinine (mg/dL)

          -  Females:

             --Creatinine CL (mL/min) = Weight (kg) x (140 - Age) x 0.85 72 x serum creatinine
             (mg/dL)

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Female subjects must be either of non-reproductive potential (i.e., post-menopausal by
             history: ≥ 50 years old and no menses for ≥1 year without an alternative medical
             cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history
             of bilateral oophorectomy) or must have a negative serum pregnancy test upon study
             entry.

          -  Willing and able to comply with the protocol for the duration of the study including
             undergoing treatment and scheduled visits and examinations with follow up

        Exclusion Criteria:

          -  Known fibrolamellar HCC, sarcomatoid HCC, or mixed cholangiocarcinoma and HCC

          -  Prior irradiation to the planned radiation target lesion

          -  Prior immunotherapy including but not limited to anti-CTLA4, including tremelimumab
             anti-PD-1, and anti-PD-L1, including durvalumab

          -  Concurrent enrollment in another study unless it is an observational (e.g.
             non-interventional) study

          -  Received live attenuated vaccines within 30 days of first dose

          -  Mean QT interval corrected for heart rate (QTc) ≥ 470 ms using Fredericia's Correction

          -  History of primary immunodeficiency

          -  History of solid organ transplantation

          -  Active or prior documented autoimmune disease within the past 2 years (NOTE: The
             following are exceptions to this criterion: Participants with vitiligo or alopecia;
             Participants with hypothyroidism (e.g. following Hashimoto syndrome) who are stable on
             hormone replacement; Participants with celiac disease controlled by diet alone:
             Participants with Grave's disease, or any chronic skin condition not requiring
             systemic treatment. Participants without active disease in the last 5 years may be
             included but only after consultation with the study physician.)

          -  Active or prior documented inflammatory bowel disease (e.g. Crohn's disease,
             ulcerative colitis)

          -  Prior other malignancy within 2 years (except for in situ disease, which is
             permissible)

          -  History of hypersensitivity to durvalumab, tremelimumab or any excipient

          -  History of (non-infectious) pneumonitis that required steroids; or evidence of
             interstitial lung disease or active, non-infectious pneumonitis

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable
             angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active
             bleeding diatheses, or psychiatric illness/social situations that would limit
             compliance with study requirements or compromise the ability of the subject to give
             written informed consent

          -  Current or prior use of immunosuppressive medication within 28 days before the first
             dose of treatment on this protocol, with the exceptions of

               -  Intranasal and inhaled corticosteroids

               -  Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of
                  prednisone or equivalent

               -  Premedication for hypersensitivity reactions (e.g. to CT contrast for scans)

          -  Active infection including tuberculosis (clinical evaluation that includes clinical
             history, physical examination and radiographic findings, and TB testing in line with
             local practice), hepatitis B (known positive HBV surface antigen (HBsAg) result),
             hepatitis C, or human immunodeficiency virus (positive HIV 1/2 antibodies). Patients
             with a past or resolved HBV infection (defined as the presence of hepatitis B core
             antibody [anti-HBc] and absence of HBsAg) are eligible. Patients positive for
             hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative
             for HCV RNA.

          -  Subjects with uncontrolled seizures

          -  Female subjects who are pregnant or breast-feeding or male or female patients of
             reproductive potential who are not employing an effective method of birth control from
             screening to 180 days after the last dose of durvalumab + tremelimumab combination
             therapy or 90 days after the last dose of durvalumab monotherapy, whichever is the
             longer time period.

          -  Any unresolved toxicity NCI CTCAE Grade ≥ 2 from previous anticancer therapy with the
             exception of alopecia, vitiligo, and the laboratory values defined in the inclusion
             criteria

          -  Patients with Grade ≥ 2 neuropathy will be evaluated on a case-by-case basis after
             consultation with the Study Physician.

             --Patients with irreversible toxicity not reasonably expected to be exacerbated by
             treatment with durvalumab or tremelimumab may be included only after consultation with
             the Study Physician.

          -  Major surgical procedure (as defined by the Investigator) within 28 days prior to the
             first dose of IP. Note: Local surgery of isolated lesions for palliative intent is
             acceptable.

          -  Brain metastases or spinal cord compression. Patients with suspected brain metastases
             at screening should have an MRI (preferred) or CT each preferably with IV contrast of
             the brain prior to study entry or brain metastases or spinal cord compression unless
             the patient is stable (asymptomatic; no evidence of new or emerging brain metastases;
             and stable and off steroids and anti-convulsants for at least 14-28 days prior to
             start of study treatment). Following radiotherapy and/or surgery of the brain
             metastases patients must wait 4 weeks following the intervention to confirm stability.