Inclusion Criteria:

• Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care
• Females of childbearing potential (FCBP)* must have a negative serum or urine pregnancy test with a sensitivity of at least 50 milli-International Units (mIU)/mL within 10 - 14 days prior to and again within 24 hours of starting lenalidomide and ixazomib and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking lenalidomide through 90 days after the last dose of study drug; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a vasectomy from the time of signing the informed consent form through 90 days after the last dose of study drug; in the event that the male patients choose to agree to practice true abstinence, this must follow the timelines detailed above; all patients assigned to the lenalidomide treatment group must be registered in and must comply with all requirements of the Revlimid Risk Evaluation and Mitigation Strategy (REMS) program
  • A female of childbearing potential is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months
• Multiple myeloma diagnosed according to standard criteria either currently or at the time of initial diagnosis
• The patient has confirmed relapsed or refractory MM
• For patients that relapse following a response to prior treatment with bortezomib or carfilzomib, six months must have elapsed since the last dose of treatment
• The patient has received 1 to 3 prior treatment regimens
• Patients must have measurable disease defined by at least 1 of the following measurements:
  • Serum M-protein ≥ 1.0 g/dL (≥ 10 g/L) for an immunoglobulin (Ig)G myeloma, ≥ 0.1 g/dL for an immunoglobulin D (IgD) myeloma or 0.5 g/dL (≥ 5g/L) for an immunoglobulin A (IgA) myeloma
  • Urine light chain ≥ 200 mg/24 hours
  • Serum free light chain ≥ 10 mg/dL provided the free light chain (FLC) ratio is abnormal
  • Patients with oligo- or non-secretory disease must have bone marrow involvement with at least 30% plasmacytosis on aspiration
• Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2
• Absolute neutrophil count (ANC) ≥ 1,000/mm³
• Platelet count ≥ 75,000/mm³; in the case that platelets are between 50,000-75,000, the patient can be enrolled if the plasma cell count in the bone marrow is superior to ≥ 50%; to meet this hematological eligibility no transfusion support and hematological growth factor are not allowed within 14 days before study enrollment
• Total bilirubin ≤ 1.5 x the upper limit of the normal range (ULN)
• Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 2.5 x ULN
• Serum creatinine ≤ 2.5 mg/dL or a calculated creatinine clearance ≥ 50 mL/min

Exclusion Criteria:

• The patient is refractory to carfilzomib or bortezomib; (refractory is defined as patients who never achieved a response and progressed while on carfilzomib or bortezomib or within 60 days of completing treatment)
• Prior treatment with any investigational proteasome inhibitor within 6 months of study entry
• Female patients who are breast feeding or have a positive serum pregnancy test during the screening period
• Failure to have fully recovered (ie, > grade 1 toxicity) from the reversible effects of prior chemotherapy
• Diarrhea > grade 1 according to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version (v)4.03
• Prior chemotherapy and/or immunotherapy within 14 days before enrollment; major surgery within 14 days before enrollment and minor surgery within 7 days prior to cycle 1 day 1
• Radiotherapy within 14 days before enrollment; if the involved field covered ≤ 5% of the bone marrow reserve, the patient may be enrolled irrespective of the end date of radiotherapy
• Central nervous system involvement
• Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment
• Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months
• Systemic treatment, within 14 days before the first dose of ixazomib, with strong cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A) inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort
• Active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive
• Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially compromise the patient's ability to understand the patient information, to give informed consent, to comply with the treatment according to this protocol or complete the study
• Diagnosed or treated for another malignancy within 2 years before study enrollment 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 complete resection
• Patient has ≥ grade 2 peripheral neuropathy or neuropathy with pain, regardless of grade that is seen on clinical examination during the screening period
• Known intolerance to immunomodulatory drugs (IMiDs)
• History of allergic reaction/hypersensitivity to any of the study medications, their analogues or excipients in the various formulations
• Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib or lenalidomide, including difficulty swallowing
• Participation in other clinical trials, including those with other investigational agents not included in this trial, such as monoclonal antibodies, within 30 days of the start of this trial and throughout the duration of this trial
• Corticosteroid doses > 10 mg/day of prednisone or equivalent within 14 days prior to cycle 1 day 1
• Autologous or allogeneic stem cell or bone marrow transplant within 3 months prior to cycle 1 day 1
• Cytotoxic therapy within 21 days prior to cycle 1 day (D) 1
• Patients that have previously been treated with ixazomib, or participated in a study with ixazomib whether treated with ixazomib or not