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.

Patients in the expansion cohort, Schedule E, must specifically have Hodgkin lymphoma, DLBCL-GCB subtype, DLBCL-non-GCB subtype, or Mantle Cell Lymphoma (MCL). Patients with multiple myeloma will no longer be evaluated as a part of this study.

Inclusion Criteria:

• Eastern Cooperative Oncology Group (ECOG) performance status 0-2.

• Patients must have 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.

  1. As of Protocol Amendment 7, patients in the expansion cohort, Schedule E, must specifically have Hodgkin lymphoma, DLBCL-GCB subtype, DLBCL-non-GCB subtype, or Mantle Cell Lymphoma (MCL). Patients with multiple myeloma will no longer be evaluated as a part of this study.

• Tumor that is evaluable by radiography, serum or urine electrophoresis (for patients with multiple myeloma), or clinical evaluations.

  1. As of Protocol Amendment 7, tumor that is evaluable by radiography or clinical evaluations.

• Suitable venous access for the conduct of blood sampling for MLN4924

• Adequate Organ Function

Exclusion Criteria:

• Systemic antineoplastic therapy within 21 days preceding first dose of study treatment, or rituximab therapy within 2 months preceding first dose of study treatment (unless there was evidence of PD since their last dose of rituximab).
• Treatment with corticosteroids within 7 days preceding first dose of study treatment.
• Prior treatment with radiation therapy involving >25% of bone marrow; Any radiotherapy within 14 days before first dose of study treatment.
• Treatment with CYP3A inducers within 14 days before the first dose of MLN4924. Moderate and strong CYP3A inhibitors and CYP3A inducers are not permitted during the study.
• Patients requiring Coumadin who cannot be switched to a low molecular weight heparin should not be considered for this study.
• Absolute neutrophil count less than 1,000/mm3; platelet count less than 75,000/mm3.
• Calculated creatinine clearance less than or equal to 50 mL/minute.