The investigators propose to evaluate the safety of drug combinations in patients with advanced gastroesophageal cancer and other gastrointestinal (GI) malignancies. Finding effective novel therapies for patients with advanced gastric cancer and other GI malignancies is an area of great unmet need. The investigators believe that modulating the tumor microenvironment with biologic agents like cabozantinib will have synergistic effect when combined with checkpoint-based immunotherapeutics like durvalumab in this patient population. This is a phase I/II, open label, multi-cohort trial looking at safety, tolerability and efficacy endpoints.

Inclusion Criteria:

1. Age ≥ 18 years

2. Eastern Cooperative Oncology Group (ECOG) Performance Status 0-1

3. Histologically confirmed diagnosis of any of the following:

   • Gastric or gastroesophageal junction adenocarcinoma

   • Esophageal adenocarcinoma
   • Colorectal adenocarcinoma (CRC)
   • Hepatocellular carcinoma (HCC)

4. Patients should have advanced (stage 4) or locally unresectable (stage III) disease.

5. Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1

6. Patients must consent to undergo a required screening/baseline biopsy procedure (and potentially another tumor biopsy at time of disease response and progression) for correlative testing.

7. Patients with gastric, gastroesophageal, or esophageal adenocarcinoma must show evidence of progression or intolerance to at least one previous standard of care systemic therapy.

8. Patients with CRC must show evidence of progression or intolerance to at least 2 previous standard of care systemic therapy. Ras wild type patients should fail epidermal growth factor receptor (EGFR) monoclonal antibody (panitumumab or cetuximab) to be eligible.

10. For cohort 3: Patients with HCC must show evidence of disease progression or intolerance to at least 1 previous systemic regimen (not more than 2 lines of prior therapy).

11. For cohort 4: Patients with HCC must be treatment naïve or show evidence of disease progression or intolerance to at least 1 previous systemic regimen (not more than 2 lines of prior therapy).

12. Patients should have known tumor results for microsatellite instability (MSI) or mismatch repair (MMR) proteins. If unknown, analysis will be obtained through local pathology lab using archival tissue if available or the baseline tumor biopsy.

13. Recovery to baseline or ≤ Grade 1 CTCAE v5.0 from toxicities related to any prior treatments unless AE(s) are clinically nonsignificant and/or stable on supportive therapy.

14. Body weight > 66 lbs (30 kg) 15. Adequate organ and marrow function. 16. Evidence of post-menopausal status or negative urinary or serum pregnancy test for female pre-menopausal patients.

17. Women of child-bearing potential and men with partners of child-bearing potential must agree to use the highly effective forms of contraception prior to study entry, for the duration of study participation, and for 180 Days post completion of therapy. Men of child-bearing potential must not donate sperm while on this study and for 180 Days after their last study treatment.

Exclusion Criteria:

1. Prior treatment with a Programmed cell death protein 1 (PD1) or (PD-L1) inhibitor, including durvalumab, or anti PD-L2 (HCC and gastric / esophageal cancer patients with prior exposure to these agents are eligible).

2. Prior treatment with cabozantinib or other Receptor for hepatocyte growth factor (MET) or Dual MET/ Hepatocyte growth factor (HGF) monoclonal antibodies or MET/HGF tyrosine kinase inhibitors (TKIs), including crizotinib, foretinib, tivantinib, rilotumumab, and onartuzumab.

3. Abdominal fistula, GI perforation, bowel obstruction, or intra-abdominal abscess within 6 months before first dose.

   * Evidence of tumor invading the GI tract, (Defined as T4 primary tumor in patients with gastric, gastroesophageal and esophageal adenocarcinoma and CRC).

   * Gastric or esophageal varices that are untreated or incompletely treated with bleeding or high risk for bleeding. Subjects treated with adequate endoscopic therapy (according to institutional standards) without any episodes of recurrent GI bleeding requiring transfusion or hospitalization for at least 6 months prior to study entry are eligible.

4. Evidence of active peptic ulcer disease, inflammatory bowel disease (eg, Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, acute pancreatitis, acute obstruction of the pancreatic duct or common bile duct, or gastric outlet obstruction.

5. Inability to swallow tablets.

6. Uncontrollable ascites or pleural effusion.

7. Cavitating pulmonary lesion(s) or known endotracheal or endobronchial disease manifestation.

8. Clinically significant hematuria, hematemesis, or hemoptysis of >0.5 tsp (2.5ml) of red blood, or other history of significant bleeding within 12 weeks.

   * Any sign indicative of pulmonary hemorrhage within 3 months.

   * Lesions invading any major blood vessels. HCC subjects with lesions invading the hepatic portal vasculature are eligible.

9. Any unresolved toxicity CTCAE Grade ≥2 from previous anticancer therapy

10. Radiotherapy treatment to more than 30% of the bone marrow or with a wide field of radiation within 4 weeks of the first dose of study drug

11. Major surgery (eg, Gastrointestinal (GI) surgery, removal or biopsy of brain metastasis) within 8 weeks before first dose of study treatment.

12. Current or prior use of immunosuppressive medication within 14 days before the first dose of durvalumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses.

13. History of allogenic organ transplantation.

14. Active or prior documented autoimmune or inflammatory disorders

15. History of active primary immunodeficiency

16. Active infection including tuberculosis, hepatitis B, hepatitis C, or human immunodeficiency virus. HCC patients with hepatitis B or C infection are allowed per protocol specific criteria.

17. Receipt of live attenuated vaccine within 30 days prior to the first dose.

18. Uncontrolled hypertension defined as sustained blood pressure (BP) > 140 millimeter of mercury (mm Hg) systolic or > 90 mm Hg diastolic despite optimal antihypertensive treatment.

19. Stroke, including transient ischemic attack (TIA), myocardial infarction (MI), or other ischemic event, or thromboembolic event (eg, deep venous thrombosis, pulmonary embolism) within 6 months before first dose. Participants with a diagnosis of deep venous thrombosis (DVT) within 6 months are allowed if stable, asymptomatic, and treated with Low Molecular Weight Heparin (LMWH) for at least 2 weeks before first dose.

20. Has untreated central nervous system (CNS) metastases and/or carcinomatous meningitis

21. Clinically significant disorders that would preclude safe study participation.

22. History of another primary malignancy except for:

    • Malignancy treated with curative intent/ resection and with no known active disease before the first dose of investigational product (IP) and of low potential risk for recurrence.
    • Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease.
    • Adequately treated carcinoma in situ without evidence of disease.

24. Concomitant anticoagulation with oral anticoagulants (e.g., warfarin, direct thrombin and Factor Xa inhibitors) or platelet inhibitors (e.g., clopidogrel). Allowed anticoagulants are the following:

1. Low-dose aspirin for cardioprotection (per local applicable guidelines) is permitted.

2. Low-dose (prophylactic) low molecular weight heparins (LMWH) are permitted.

3. Anticoagulation with therapeutic doses of LMWH is allowed in subjects with no known brain metastases, clinically significant hemorrhage, or complications from a thromboembolic event on the anticoagulation regimen (subjects with HCC must also have a screening platelet count >100,000/μl), and who have been on a stable dose of LMWH for at least 1 week before first dose.

   25. Known allergy or hypersensitivity to any of the study drugs or any of the study drug excipients.