Study of Ixazomib and Romidepsin in Peripheral T-cell Lymphoma (PTCL)

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Age ≥ 18 years at the time of consent. * ECOG Performance Status of 0-2 within 14 days prior to registration. * Histological confirmation of peripheral T-cell lymphoma (PTCL) and biopsy confirmation of disease relapse (after initial or any subsequent salvage therapy). * Documented disease progression after receiving at least one prior therapeutic regimen. * Prior cancer treatment must be completed at least 28 days prior to registration and the subject must have recovered from all reversible acute toxic effects of the regimen (other than alopecia) to ≤ Grade 1 or baseline. Systemic steroids at a dose less than the equivalent of 10 mg/day of prednisone and inhaled, nasal, and topical steroids are permitted. Intermittent dexamethasone for the treatment of nausea/emesis is also permitted. * Absolute Neutrophil Count (ANC) ≥ 1000/mm3 * Platelets (Plt) ≥ 75,000/mm3 * Calculated creatinine clearance ≥ 30 cc/min using the Cockcroft-Gault formula * Bilirubin ≤ 1.5 × upper limit of normal (ULN), (exception of Gilbert disease) * Aspartate aminotransferase (AST) ≤ 3 × ULN, if known hepatic involvement then ≤ 5 × ULN * Alanine aminotransferase (ALT) ≤ 3 × ULN, if known hepatic involvement then ≤ 5 × ULN * Females of childbearing potential must have a negative serum pregnancy test within 14 days prior to registration. NOTE: Females are considered of child bearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months. * Males must be willing to abstain from donating sperm or semen from the time of informed consent until 90 days after treatment discontinuation. * The subject must have the ability to understand and comply with study procedures for the entire length of the study, as determined by the treating physician or protocol designee. Exclusion Criteria: * A history of, or a concurrent, clinically significant illness, medical condition or laboratory abnormality that, in the investigator's opinion, could affect the conduct of the study. * Active infection requiring systemic therapy * Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study). * Known additional malignancy that is active and/or progressive requiring treatment; exceptions include basal cell or squamous cell skin cancer, in situ cervical or bladder cancer, or other cancer for which the subject has been disease-free for at least two years. * Active central nervous system (CNS) lymphoma * Major surgery or radiation therapy within 28 days of study registration * Uncontrolled infectious disease, including active herpes simplex or herpes zoster * Known positive test for Hepatitis B surface antigen, Hepatitis C, or HIV. NOTE: testing is not required. * Known GI disease or GI procedure that could interfere with the oral absorption or tolerance of oral medications including difficulty swallowing, as determined by the treating physician. * Evidence of uncontrolled cardiovascular conditions, including uncontrolled hypertension, cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months. * Q-T interval, based on Bazett-corrected interval \> 0.45 sec * Treatment with any investigational drug within 28 days prior to registration. * Peripheral neuropathy ≥ grade 2 * Prior treatment with bortezomib, ixazomib, or romidepsin. * Systemic treatment, within 14 days, with strong inhibitors of CYP1A2 (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of CYP3A (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of St. John's wort. * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol. * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent. * Prior autologous hematopoietic stem cell transplant within 90 days of study registration. * Prior allogeneic hematopoietic stem cell transplant.