The main purpose of this study is to see if regorafenib can help control or decrease cancer size in patients with cancer of the bile duct. Researchers also want to find out if regorafenib is safe and tolerable.

Inclusion Criteria:

• Histologically or cytologically documented carcinoma primary to the intra- or extra-hepatic biliary system or gall bladder with clinical and/or radiologic evidence of unresectable, locally advanced or metastatic disease. Patients with ampullary carcinoma are not eligible.
• Have failed no more than 2 prior lines of systemic chemotherapy for advanced biliary cancer. Patients who received adjuvant chemotherapy and had evidence of disease recurrence within 6 months of completion of the adjuvant treatment are also eligible. If patient received adjuvant treatment and had disease recurrence after 6 months, they will only be eligible after failing one line of systemic chemotherapy used to treat the disease recurrence.
• Eastern Cooperative Oncology Group (ECOG) Performance Status Assessment of 0 or 1
• Measurable and non-measurable disease will be allowed.
• Must not have been treated with any vascular endothelial growth factor (VEGF) inhibitors. Prior 5-Fluorouracil (5-FU) or capecitabine treatment is allowed only if given as a radiosensitizer concurrently with radiation therapy at least 12 weeks prior to registration or if given as part of any adjuvant therapy regimen > 6 months prior to study enrollment.
• Life expectancy of at least 12 weeks (3 months)
• For patients who have received prior cryotherapy, radiofrequency ablation, therasphere, ethanol injection, transarterial chemoembolization (TACE) or photodynamic therapy, the following criteria must be met: 28 days have elapsed since that therapy (lesions that have not been treated with local therapy must be present and measureable.
• Able to understand and willing to sign the written informed consent form
• All acute toxic effects of any prior treatment have resolved to National Cancer Institute Common Terminology Criteria for Adverse Events (NCI-CTCAE) v4.0 Grade 1 or less at the time of signing the Informed Consent Form (ICF).
• Adequate bone marrow and liver function
• Participants can receive 5-FU or capecitabine.
• Women of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of study drug.
• Men and women of childbearing potential must agree to use adequate contraception beginning at the signing of the ICF until at least 3 months after the last dose of study drug.
• Able to swallow and retain oral medication

Exclusion Criteria:

• Previous assignment to treatment during this study. Participants permanently withdrawn from study participation will not be allowed to re-enter study.
• Other investigational treatment during or within 21 days before starting study treatment
• Child Pugh B and C
• Uncontrolled hypertension (systolic pressure >140 mm Hg or diastolic pressure > 90 mm Hg [NCI-CTCAE v4.0] on repeated measurement) despite optimal medical management
• Active or clinically significant cardiac disease
• Evidence or history of bleeding diathesis or coagulopathy
• Any hemorrhage or bleeding event ≥ NCI CTCAE Grade 3 within 4 weeks prior to start of study medication
• Participants with thrombotic, embolic, venous, or arterial events, such as cerebrovascular accident (including transient ischemic attacks) deep vein thrombosis or pulmonary embolism within 6 months of informed consent
• Active malignancy except for nonmelanoma skin cancer or in situ cervical cancer. Potential participants surviving a cancer that was curatively treated and without evidence of disease for more than 3 years before the trial are allowed. All cancer treatments must be completed at least 3 years prior to study entry (i.e., signature date of the informed consent form).
• Potential participants with phaeochromocytoma
• Potential participants with severe hepatic impairment (Child-Pugh Class C)
• Known history of human immunodeficiency virus (HIV) infection or current chronic or active hepatitis B or C infection requiring treatment with antiviral therapy.
• Ongoing infection > Grade 2 NCI-CTCAE v4.0
• Symptomatic metastatic brain or meningeal tumors
• Presence of a non-healing wound, non-healing ulcer, or bone fracture
• Renal failure requiring hemo-or peritoneal dialysis
• Patients with seizure disorder requiring medication
• Persistent proteinuria >/= Grade 3 NCI-CTCAE v4.0 (> 3.5 g/24 hours, measured by urine protein:creatinine ratio on a random urine sample)
• Interstitial lung disease with ongoing signs and symptoms at the time of informed consent
• Pleural effusion or ascites that causes respiratory compromise (≥ NCI-CTCAE version 4.0 Grade 2 dyspnea)
• History of organ allograft (including corneal transplant)
• Known or suspected allergy or hypersensitivity to any of the study drugs, study drug classes, or excipients of the formulations given during the course of this trial
• Any malabsorption condition
• Women who are pregnant or breast-feeding
• Any condition which, in the investigator's opinion, makes the potential participant unsuitable for trial participation
• Substance abuse, medical, psychological or social conditions that may interfere with participation in the study or evaluation of the study results

From consent to study completion, 3 years 3 months