A Study of BBI503 in Adult Patients With Advanced Urologic Malignancies

Inclusion Criteria: * Signed written informed consent * Histologically or cytologically confirmed renal cell cancer or urothelial carcinoma that is metastatic, unresectable, or recurrent. * ≥ 18 years of age * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Male or female patients of child-producing potential must agree to use contraception or avoidance of pregnancy measures during the study and for 30 days after the last BBI503 dose * Females of childbearing potential must have a negative serum pregnancy test * Alanine transaminase (ALT) ≤ 2.5 x the upper limit of normal (ULN), or ≤ 3.5 x ULN in the presence of primary or metastatic hepatic lesions * Hemoglobin (Hgb) ≥ 10 g/dl * Total bilirubin ≤ 1.5 x ULN * Creatinine ≤ 1.5 x ULN or creatinine clearance \> 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal. * Absolute neutrophil count ≥ 1.5 x 10\^9/L * Platelets ≥ 100 x 10\^9/L * Prothrombin time (PT) ≤ 16 seconds and partial thromboplastin time (PTT) ≤ 1.5 x ULN * Life expectancy ≥ 3 months * A patient with renal cell carcinoma (RCC) must be classified into clear-cell or non-clear cell subtype, and must then meet the following criteria: * Patients with clear cell renal cancer must have received prior treatment with at least one line of standard systemic therapy which included treatment with a tyrosine kinase inhibitor; either alone or in combination. The patient must have had either progressive disease during treatment, or have had documented intolerance to first line tyrosine kinase inhibitor (TKI)-based therapy. * Patients with clear cell renal cancer who have received anti-cancer treatment beyond first-line TKI-based therapy may enroll. * Patients with clear cell renal cancer who have not received anti-cancer treatment beyond first-line TKI-based therapy may enroll only after discussing the potential risks and benefits of alternative treatment options, such as other currently available agents approved for treatment of patients with clear cell renal cancer. * Patients with non-clear cell RCC which is metastatic may enroll without having received prior treatment in the metastatic setting * A patient with urothelial carcinoma must also meet the following criteria: * Must have received at least one prior line of standard systemic cytotoxic therapy in the metastatic setting. Exclusion Criteria: * Anti-cancer chemotherapy, radiotherapy, immunotherapy, or investigational agents within 7 days of first dose of BBI503. Patients may begin BBI503 on a date determined by the investigator and medical monitor for the sponsor provided there is a minimum of 7 days since last receiving anti-cancer treatment, and that all prior treatment-related adverse events (AEs) have resolved or have been deemed irreversible. * Major surgery within 4 weeks prior to first dose (requiring general anesthesia and/or inpatient hospitalization for recovery). * Any known symptomatic or untreated brain metastases requiring increase of steroid dose within 2 weeks prior to starting on study. Patients with treated brain metastases must be stable for 4 weeks after completion of that treatment. Post-treatment image documentation of stability is required within 4 months of starting on study. Patients must have no clinical symptoms from brain metastases and must be either off steroids or on a stable dose of steroids for at least 2 weeks prior to protocol enrollment. Patients with known leptomeningeal metastases are excluded, even if treated. * Pregnant or breastfeeding * Significant gastrointestinal disorder(s), in the opinion of the Principal Investigator, (e.g., Crohn's disease, ulcerative colitis, extensive gastric and small intestine resection) * Unable or unwilling to swallow BBI503 capsules daily * Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, clinically significant non-healing or healing wounds, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, significant pulmonary disease (shortness of breath at rest or mild exertion), uncontrolled infection or psychiatric illness/social situations that would limit compliance with study requirements (e.g. no reliable transportation). * Patients with a history of malignancies other than the tumor of interest except: adequately treated non-melanoma skin cancer, curatively treated in-situ cancer of the cervix, or other solid tumors curatively treated with no evidence of disease for \> 3 years. * Abnormal ECGs which are clinically significant such as QT prolongation - QTc \> 480 msec, clinically significant cardiac enlargement or hypertrophy, new bundle branch block, or signs of active ischemia. Patients with evidence of prior infarction who are New York Heart Association (NYHA) functional class II, III, or IV are excluded, as are patients with marked arrhythmia such as Wolff Parkinson White pattern or complete atrioventricular (AV) dissociation.