A Study of Ixazomib and Ibrutinib in Relapsed/Refractory Mantle Cell Lymphoma

* Relapsed or refractory, pathologically proven mantle cell lymphoma. Must have a current or prior tissue sample that is IHC positive for cyclin D 1 or that is positive by FISH or cytogenetics for t(11;14). * Must have been refractory to and/or relapsed/progressed after at least 1 prior therapy. * Prior autologous or allogeneic transplant are allowed. Patients may not have active grade II-IV acute graft-versus-host disease (GVHD) or moderate/severe chronic GVHD by NIH criteria and may not require immunosuppressive medications and/or corticosteroids for the management of acute or chronic GVHD. * Phase I: Prior proteasome inhibitor and/or Bruton's tyrosine kinase (BTK) inhibitors are allowed but patients may not have been exposed to the combination of proteasome inhibitor and BTK inhibitor. Patients who have progressed on ibrutinib that are felt to be at high risk for rapid progression on this study shall not be eligible for the phase I portion of the study. NOTE: Ibrutinib pre-treated patients must meet all eligibility criteria AND must have discontinued prior ibrutinib at least 3 months prior to starting study therapy. PHASE I COMPLETED NOVEMBER 25, 2019. * Phase II: Prior proteasome inhibitors allowed. (Please note prior to Version 3.0 of the protocol prior proteasome inhibitor and/or Bruton's tyrosine kinase inhibitors were allowed but patients could not have been exposed to the combination of proteasome inhibitor and BTK inhibitor). * Age ≥ 18 years. * Eastern Oncology Oncology Group (ECOG) performance status of 0-2. * Ability to understand and willingness to sign Institutional Review Board (IRB)-approved informed consent. * Willing to provide archived tumor tissue, bone marrow (if sufficient bone marrow and tumor tissue are available) and blood samples for research. * Adequate organ function as measured by the following criteria * Absolute Neutrophil Count (ANC) ≥ 750/mm³ * Platelets ˃50,000/mm³ * Serum Creatinine ≤ 2x Upper Limit Normal (ULN) * ALT and AST ≤ 3x ULN * Total Bilirubin ≤ 1.5x ULN * Patients must not have received systemic treatment for MCL for at least 14 days prior to enrollment, except for steroids which may be used to manage acute symptoms related to disease up to 48 hours prior to starting study therapy. Radiation therapy must be concluded at least 14 days prior to enrollment. * Women must not be pregnant or breastfeeding since we do not know the effects of ixazomib and ibrutinib on the fetus or breastfeeding child. All sexually active females of childbearing potential must have a blood test to rule out pregnancy within 2 weeks prior to registration. * Sexually active women of child-bearing potential with a non-sterilized male partner and sexually active men must agree to use 2 methods of adequate contraception (hormonal plus barrier or 2 barrier forms) OR abstinence prior to study entry, for the duration of study participation, and for 3 months following last dose of study drugs. * Patients must have resolved all prior non-hematologic toxicities assessed as related to prior therapy to ≤ grade 1. * Patients must have measurable disease (i.e., ≥ 1.5 cm in largest diameter) by conventional imaging modalities. Patients with extranodal involvement as the only measurable site of disease must have a largest diameter ≥ 1.0 cm and must be attributable to active lymphoma in the opinion of the investigator. * Patients may not have current/active Central Nervous System (CNS) involvement with mantle cell lymphoma (patients with prior CNS involvement are eligible as long as they have had no evidence of active CNS disease for at least 6 months). * Patients may not have another malignancy that could interfere with the evaluation of safety or efficacy of this combination. Patients with a prior malignancy will be allowed without study chair approval in the following circumstances: * Not currently active and diagnosed at least 3 years prior to the date of enrollment. * Non-invasive diseases such as low risk cervical cancer or any cancer in situ * Localized disease in which chemotherapy would not be indicated (such as Stage I colon, lung, prostate or breast cancer). Patients with other malignancies not meeting these criteria must be discussed with PrECOG prior to enrollment. * Patients requiring long-term anticoagulation must be managed on an anticoagulant besides warfarin. Patients who require warfarin are not eligible. * Patients with a clinically significant bleeding episode as judged by the investigator within 3 months of registration are not eligible, except patients who suffer bleeding due to trauma. * Patients may not have had major surgery within 14 days, or minor surgery within 3 days, before registration. * Patients may not have any active infection requiring oral or intravenous antimicrobial therapy at the time of therapy initiation. Patients with a recent self-limited infection that has clinically resolved may complete a prescribed course of antimicrobial therapy after study initiation as long as they are asymptomatic with no clinical evidence of infection for at least 7 days prior to treatment. Patients with a recent serious (grade ≥ 3) infection requiring hospitalization must have completed all antimicrobial therapy within 14 days of therapy initiation. * Patients may not have evidence of uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure (New York Heart Association (NYHA) class III or higher, unstable angina, or myocardial infarction within the past 6 months. Patients with a history of any significant cardiovascular disease that has been controlled for at least 14 days before registration are allowed (except for patients who have had a myocardial infarction within 6 months). * No systemic treatment, within 14 days before the first dose of ibrutinib with moderate or strong inhibitors of CYP3A (Strong Inhibitors: ketoconazole, itraconazole, voriconazole, posaconazole, clarithromycin, and telithromycin; Moderate Inhibitors: fluconazole, darunavir, erythromycin, diltiazem, atazanavir, aprepitant, amprenavir, fosamprenavir, crizotinib, imatinib, verapamil, ciprofloxacin, grapefruit juice products, and Seville oranges) or strong CYP3A inducers for ibrutinib and ixazomib (carbamazepine, rifampin, phenytoin, St. John's wort). * Patients with ongoing or active systemic infection, active hepatitis B or C virus infection, or known Human Immunodeficiency Virus (HIV) positive are not eligible. Testing is not required in absence of clinical suspicion. * Patients with a history of hepatitis B or C must have a negative peripheral blood Polymerase Chain Reaction (PCR) and may not be positive for Hepatitis B surface antigen. Patients with cirrhosis or other evidence of liver damage due to Hepatitis B or C are not eligible. * Patients with any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of the treatment according to the protocol are not eligible. * Patients with a known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent are not eligible. * Patients with known gastrointestinal (GI) disease or prior GI procedure that could interfere with the oral absorption or tolerance of ixazomib or ibrutinib including difficulty swallowing are not eligible. * Patients with ≥ Grade 2 peripheral neuropathy, or Grade 1 peripheral neuropathy with pain on clinical examination during the screening period are not eligible. * Patients may not participate in any other therapeutic clinical trials, including those with other investigational agents not included in this trial throughout the duration of this study. * As ibrutinib will not be provided by the study, the patient must be able to obtain ibrutinib through other means (i.e., commercially or through patient assistance programs). This must be confirmed prior to registration.