Inclusion Criteria:

          -  Phase 1: Relapsed MM with at least one prior line of therapy and should have received
             a proteasome inhibitor and an immunomodulatory drug

          -  Phase 2: 1-3 prior lines of therapy and should have received a proteasome inhibitor
             and an immunomodulatory drug

          -  Obtained =< 14 days prior to registration: Calculated creatinine clearance (using
             Cockcroft-Gault equation) >= 30 mL/min

          -  Obtained =< 14 days prior to registration: Absolute neutrophil count (ANC) >= 1000/uL
             (without growth factor support)

          -  Obtained =< 14 days prior to registration: Un-transfused platelet count >= 75000/uL
             (>= 50,000/uL if marrow plasma cells [PC]% > 50%)

          -  Obtained =< 14 days prior to registration: Hemoglobin >= 8.0 g/dL

          -  Obtained =< 14 days prior to registration: Total bilirubin =< 1.5 x upper limit of
             normal (ULN)

          -  Obtained =< 14 days prior to registration: Alanine aminotransferase (ALT) and
             aspartate aminotransferase (AST) =< 3 x ULN

          -  Obtained =< 14 days prior to registration: Alkaline phosphatase =< 750 U/L

          -  Expansion cohort only: Plasma cell fluorescence in situ hybridization (FISH) test
             demonstrating presence of t(11;14)

          -  Measurable disease of multiple myeloma as defined by at least ONE of the following:

               -  Serum monoclonal protein >= 1.0 g/dL

               -  > 200 mg of monoclonal protein in the urine on 24 hour electrophoresis

               -  Serum immunoglobulin free light chain >= 10 mg/dL AND abnormal serum
                  immunoglobulin kappa to lambda free light chain ratio

          -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2

          -  Provide written informed consent

          -  Negative serum pregnancy test done =< 7 days prior to registration, for women of
             childbearing potential only

          -  Willing to follow strict birth control measures as suggested by the study

               -  Female patients: If they are of childbearing potential, must agree to one of the
                  following:

                    -  Practice 2 effective methods of contraception, at the same time, from the
                       time of signing the informed consent form through 90 days after the last
                       dose of study drug, AND must also adhere to the guidelines of any
                       treatment-specific pregnancy prevention program, if applicable, OR

               -  Agree to practice true abstinence when this is in line with the preferred and
                  usual lifestyle of the subject; (periodic abstinence [e.g., calendar, ovulation,
                  symptothermal, post-ovulation methods] and withdrawal are not acceptable methods
                  of contraception)

               -  Male patients: even if surgically sterilized (i.e., status post-vasectomy), must
                  agree to one of the following:

                    -  Agree to practice effective barrier contraception during the entire study
                       treatment period and through 120 days after the last dose of study drug, OR

                    -  Agree to practice true abstinence when this is in line with the preferred
                       and usual lifestyle of the subject; (periodic abstinence (e.g., calendar,
                       ovulation, symptothermal, post-ovulation methods and withdrawal are not
                       acceptable methods of contraception)

          -  Life expectancy >= 12 weeks

          -  Willing to return to enrolling institution for follow-up (during the Active Monitoring
             Phase of the study)

          -  Willing to provide research bone marrow aspirate specimen

        Exclusion Criteria:

          -  Diagnosed or treated for another malignancy =< 2 years before study enrollment or
             previously diagnosed with another malignancy and have any evidence of residual
             disease; NOTE: Patients with non-melanoma skin cancer or carcinoma in situ of any type
             are not excluded if they have undergone complete resection

          -  Any of the following:

               -  Pregnant women

               -  Nursing women

               -  Men or women of childbearing potential who are unwilling to employ adequate
                  contraception

          -  Other co-morbidity which would interfere with patient's ability to participate in
             trial, e.g. uncontrolled infection, uncompensated heart or lung disease

          -  Other concurrent chemotherapy or any ancillary therapy considered investigational

               -  NOTE: Bisphosphonates are considered to be supportive care rather than therapy,
                  and are thus allowed while on protocol treatment

          -  Peripheral neuropathy >= grade 2 on clinical examination or grade 1 with pain during
             the screening period

          -  Major surgery =< 14 days prior to study registration

          -  History or current evidence of any condition, therapy, or laboratory abnormality that
             might confound the results of the trial, interfere with the subject?s participation
             for the full duration of the trial, or is not in the best interest of the subject to
             participate, in the opinion of the treating investigator

          -  Known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies) or active
             hepatitis

          -  Administration of a strong or moderate CYP3A inhibitor or inducer =< 14 days prior to
             registration

          -  Known allergy to any of the study medications, their analogues, or excipients in the
             various formulations of any agent

          -  Participation in other clinical trials, including those with other investigational
             agents not included in this trial, within 30 days of the start of this trial and
             throughout the duration of this trial

          -  Failure to have fully recovered (i.e., =< grade 1 toxicity) from the reversible
             effects of prior chemotherapy

          -  Radiotherapy =< 14 days prior to registration; Note: If the involved field is small, 7
             days will be considered a sufficient interval between treatment and administration of
             the ixazomib

          -  Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of ixazomib including difficulty swallowing

          -  Previous treatment with ixazomib, or participated in a blinded study with ixazomib

          -  Live-virus vaccines =< 28 days prior to registration

          -  Heart failure > New York Heart Association (NYHA) class II