A Dose Finding Study Followed by a Safety and Efficacy Study for Patients With Multiple Myeloma or Lymphoma

Inclusion Criteria: 1. Provide written informed consent 2. Women of child-bearing potential must have a negative pregnancy test Multiple Myeloma: Confirmed diagnosis of Multiple Myeloma with measurable disease, having been treated with at least two prior lines of therapy. Lymphoma: Confirmed diagnosis of Non-Hodgkin Lymphoma with measurable disease, having been treated with at least one anthracycline-based therapy, with relapse or progression since the last treatment received. Exclusion Criteria: 1. Any of the following treatments, within the specified time frame, prior to the first dose of TAS4464: 1. Major surgery within 28 days 2. Radiation/chemotherapy within 21 days 3. Monoclonal antibodies within 28 days 4. Corticosteroid administration \>20 mg/day of prednisone or equivalent within 14 days 5. Proteasome inhibitors within 14 days 6. Immunomodulatory agents within 7 days 7. Stem cell transplant within 3 months 8. Current immunosuppressive treatment for graft versus host disease 9. Current use of an investigational agent 2. Active graft versus host disease 3. Known serious illness or medical condition 4. Prior treatment with TAS4464 or known hypersensitivity to any of its inactive ingredients or drugs similar in class 5. Pregnant or breast-feeding female