This is an open-label, multicenter, phase 1, dose escalation study of MLN4924 in adult patients with lymphoma or multiple myeloma. The patient population will consist of adults with a confirmed diagnosis of lymphoma (Waldenstrom's macroglobulinemia is permitted) or multiple myeloma that is relapsed and/or refractory after at least 2 prior standard chemotherapeutic regimens and for which no curative option exists.