Inclusion Criteria:

          -  Patients must have RRMM with measurable disease, as defined by at least one of the
             following:

               -  Serum monoclonal protein >= 0.5 g/dL

               -  Urinary monoclonal protein excretion of >= 200 mg/24 hours

               -  Kappa or lambda light chain level >= 10 mg/dL with an abnormal free light chain
                  ratio

          -  At least two prior lines of therapy and all patients should have at least been exposed
             to a proteasome inhibitor (PI), an immunomodulatory drug (IMiD), and an anti-CD38
             antibody.

               -  For proteasome-sensitive expansion cohort: Patients with MM who relapsed or are
                  refractory to a prior line of therapy not including a proteasome inhibitor

               -  For proteasome-relapsed/refractory expansion cohort: Patients with MM who have
                  relapsed after prior PI exposure or are PI-refractory, defined as nonresponsive
                  to treatment or progresses within 60 days of last exposure to a PI

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,000/mcL

          -  Platelets >= 75,000/mcL

          -  Bilirubin =< institutional upper limit of normal (ULN).

               -  Patients with Gilbert's syndrome may enroll if direct bilirubin =< 1.5 x ULN

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 3 x institutional ULN

          -  Creatinine clearance (CrCl) by Cockcroft-Gault >= 30 mL/min

          -  Hemoglobin > 8 g/dL. Patients may be transfused to achieve this value. Elevated
             indirect bilirubin due to post-transfusion hemolysis is allowed

          -  Known human immunodeficiency virus (HIV) positive patients who meet the following
             criteria will be considered eligible:

               -  CD 4 count > 350 cells/mm^3

               -  Undetectable viral load

               -  Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents
                  (e.g. excluding ritonavir)

               -  No history of acquired immune deficiency syndrome (AIDS)-defining opportunistic
                  infections

          -  The effects of MLN4924 (pevonedistat) and MLN9708 (ixazomib) on the developing human
             fetus are unknown. For this reason and because NAE inhibitory agents are known to be
             teratogenic, women of child-bearing potential and men must meet the following
             criteria:

               -  Female patients who are:

                    -  Postmenopausal for at least one year before the screening visit, OR

                    -  Surgically sterile, OR

                    -  If of childbearing potential, agree to practice 1 highly effective method
                       and 1 additional (barrier) method of contraception, at the same time, from
                       the time of signing the informed consent until 4 months after the last dose
                       of the ixazomib and pevonedistat (female and male condoms should not be used
                       together), or agree to abstain from heterosexual intercourse, when this is
                       in line with the preferred and usual lifestyle of the subject (Periodic
                       abstinence [e.g,, calendar, ovulation, symptothermal, postovulation methods]
                       withdrawal, spermicides only, and lactational amenorrhea are not acceptable
                       methods of contraception)

               -  Male patients, even if surgically sterilized, who:

                    -  Agree to practice effective barrier contraception during the entire time
                       enrolled on study through 4 months after completion of ixazomib and
                       pevonedistat administration (female and male condoms should not be used
                       together), OR

                    -  Agree to abstain from heterosexual intercourse, when this is in line with
                       the preferred and usual lifestyle of the subject. (Periodic abstinence
                       [e.g., calendar, ovulation, symptothermal, postovulation methods for the
                       female partner] withdrawal, spermicides only, and lactational amenorrhea are
                       not acceptable methods of contraception)

               -  Should a woman become pregnant or suspect she is pregnant while she or her
                  partner is participating in this study, she should inform her treating physician
                  immediately

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible

        Exclusion Criteria:

          -  Diagnosed or treated for another malignancy within 2 years before randomization or
             previously diagnosed with another malignancy and have any evidence of residual
             disease. Patients with non-melanoma skin cancer or carcinoma in situ of any type are
             not excluded if they have undergone resection

          -  Patients who are receiving any other investigational agents, within 30 days of the
             start of this trial and throughout the duration of this trial

          -  Patients with known central nervous system involvement should be excluded from this
             clinical trial because of their poor prognosis and because they often develop
             progressive neurologic dysfunction that would confound the evaluation of neurologic
             and other adverse events (AEs)

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to MLN4924 (pevonedistat) or MLN9708 (ixazomib) (including boron or
             boron-containing products)

          -  Patients with uncontrolled intercurrent illness

          -  Pregnant women are excluded from this study because MLN4924 (pevonedistat) is an NAE
             inhibitory agent with the potential for teratogenic or abortifacient effects. Because
             there is an unknown but potential risk for AEs in nursing infants secondary to
             treatment of the mother with MLN4924 (pevonedistat), breastfeeding should be
             discontinued if the mother is treated with MLN4924 (pevonedistat). These potential
             risks may also apply to the use of MLN9708 (ixazomib) in this study

          -  Major surgery within 14 days before the first dose of any study drug or a scheduled
             surgery during study period

          -  Patients with uncontrolled coagulopathy or bleeding disorder

          -  Known hepatic impairment as defined by known hepatic cirrhosis, hepatitis B surface
             antigen seropositive or known or suspected active hepatitis C infection

               -  Note: Patients who have isolated positive hepatitis B core antibody (i.e., in the
                  setting of negative hepatitis B surface antigen and negative hepatitis B surface
                  antibody) must have an undetectable hepatitis B viral load. Patients who have
                  positive hepatitis C antibody may be included if they have an undetectable
                  hepatitis C viral load

          -  Known cardiopulmonary disease defined as:

               -  Unstable angina;

               -  Congestive heart failure (New York Heart Association [NYHA] class III or IV);

               -  Myocardial infarction within 6 months prior to first dose (patients who had
                  ischemic heart disease such as acute coronary syndrome [ACS], myocardial
                  infarction, and/or revascularization greater than 6 months before screening and
                  who are without cardiac symptoms may enroll);

               -  Symptomatic cardiomyopathy

               -  Clinically significant arrhythmia:

                    -  History of polymorphic ventricular fibrillation or torsade de pointes,

                    -  Permanent atrial fibrillation, defined as continuous atrial fibrillation for
                       >= 6 months,

                    -  Persistent atrial fibrillation, defined as sustained atrial fibrillation
                       lasting > 7 days and/or requiring cardioversion in the 4 weeks before
                       screening,

                    -  Grade 3 atrial fibrillation defined as symptomatic and incompletely
                       controlled medically, or controlled with device (e.g., pacemaker), or
                       ablation and

                    -  Patients with paroxysmal atrial fibrillation or grade < 3 atrial
                       fibrillation for period of at least 6 months are permitted to enroll
                       provided that their rate is controlled on a stable regimen

               -  Symptomatic pulmonary hypertension

          -  Uncontrolled high blood pressure (i.e., systolic blood pressure > 180 mmHg, diastolic
             blood pressure > 95 mmHg)

          -  Prolonged rate corrected QT (QTc) interval >= 500 msec, calculated according to
             institutional guidelines

          -  Left ventricular ejection fraction (LVEF) < 50% as assessed by echocardiogram

          -  Known moderate to severe chronic obstructive pulmonary disease, interstitial lung
             disease, and pulmonary fibrosis

          -  Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of MLN9708 (ixazomib), including difficulty swallowing

          -  Peripheral neuropathy that is grade >= 3, or grade 2 with pain on clinical examination
             during the screening period

          -  Patients that have previously been treated with MLN9708 (ixazomib)

          -  Systemic treatment, within 14 days before the first dose of MLN9708 (ixazomib), with
             strong CYP3A inducers (rifampin, rifapentine, rifabutin, ritonavir, carbamazepine,
             phenytoin, phenobarbital), or use of St. John's wort. Clinically significant metabolic
             enzyme inducers are not permitted during this study

          -  Radiotherapy within 14 days before enrollment. If the involved field is small, 7 days
             will be considered a sufficient interval between treatment and administration of the
             MLN9708 (ixazomib)

          -  Female patients who intend to donate eggs (ova) during the course of this study or 4
             months after receiving their last dose of study drug(s)

          -  Male patients who intend to donate sperm during the course of this study or 4 months
             after receiving their last dose of study drug(s)