The Purpose of This Trial is to Determine if Regorafenib Plus Durvalumab (MEDI4736) is Safe and Effective in Treatment of Chemo Refractory Advanced Biliary Tract Cancers

Inclusion Criteria: * Ability of patient OR Legally Authorized Representative (LAR) to understand this study, and participant or LAR willingness to sign a written informed consent * Can swallow tablets and self-administer medication * Progressed on at least one line of therapy (no restrictions on type of previous treatment) * Eastern Cooperative Oncology Group (ECOG) Performance Status = 0 - 1 * Measurable disease with at least 1 lesion that qualifies as a RECIST 1.1 Target Lesion (TL) at baseline. Previously irradiated lesion cannot be considered as Target Lesion (TL) except in cases of documented progression of the lesion since the completion of radiation therapy * Histologically confirmed unresectable or metastatic intrahepatic/extrahepatic cholangiocarcinoma or gallbladder cancer with radiographic progression, who have progressed on one line of therapy / failed adjuvant therapy * Life expectancy of at least 3 months * Recovery to baseline or \< Grade 2 CTCAE v5.0 from toxicities related to any prior treatments, unless adverse event's (AE(s)) are clinically nonsignificant and/or stable on supportive therapy * Adequate organ function per laboratory results * Concomitant anticoagulation with oral anticoagulants (e.g., warfarin, direct thrombin and Factor Xa inhibitors) or platelet inhibitors (e.g., clopidogrel) will be allowed provided that no prior evidence of underlying abnormality in coagulation parameters exists. Close monitoring of at least weekly evaluations will be p…