A Phase 2 Multicenter Study of High Dose Chemotherapy With Autologous Stem Cell Transplant Followed by Maintenance Therapy With Romidepsin for the Treatment of T Cell Non-Hodgkin Lymphoma

Inclusion Criteria: * Age: Patients over age 16 who are deemed eligible for transplant by their treating physician Disease status: CR or PR required. Remission status will be assessed at the completion of induction chemotherapy and prior to enrollment on protocol. Diagnosis: The following histologies will need to be confirmed at MSK or locally for participating sites in order to be considered for HDT-ASCT and post-transplant maintenance romidepsin: * PTCL * AITL * ALCL * EaTCL * Hepatosplenic Gamma Delta T cell lymphoma * Adult T-cell leukemia/lymphoma * Primary cutaneous gamma/delta T-cell lymphoma * Extranodal NK/T-cell lymphoma, nasal type * Primary cutaneous anaplastic large cell lymphoma * Subcutaneous panniculitis-like T-cell lymphoma * Mycosis fungoides/sezary syndrome Stem cell collection: A minimum of 2 x 106 CD34+ cells must have been collected Laboratory test results within these ranges: * Total bilirubin \<= 1.5 x ULN * AST (SGOT) and ALT (SGPT) \<= 3 x ULN Exclusion Criteria: * Diagnosis: progressive disease at transplant work-up * Prior therapy: prior autologous or allogeneic transplant * Active and uncontrolled infection at time of transplantation including active infection with Aspergillus or other mold, or HIV infection * Inadequate performance status/organ function defined by DLCO \< 50% (adjusted for hgb), cardiac function as defined below, KPS \< 60%. * Pregnant or breast feeding. For males and females of child-producing potential, inability to use effective contraceptive methods during the study * Prior therapy with romidepsin * Central nervous system or meningeal involvement * Any known cardiac abnormalities such as: * Congenital long QT syndrome * QTc interval ≥ 500 milliseconds * Myocardial infarction within 6 months of transplantation. Subjects with a history of myocardial infarction between 6 and 12 months prior to transplant who are asymptomatic and have had a negative cardiac risk assessment (treadmill stress test, nuclear medicine stress test, or stress echocardiogram) since the event may participate * Other significant ECG abnormalities including 2nd degree atrio-ventricular (AV) block type II, 3rd degree AV block, or bradycardia (ventricular rate less than 50 beats/min) * Symptomatic coronary artery disease (CAD), e.g., angina Canadian Class II-IV (see Appendix 1) In any patient in whom there is doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present * An ECG recorded at screening showing evidence of cardiac ischemia (ST depression of ≥2 mm, measured from isoelectric line to the ST segment). If in any doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present * Congestive heart failure (CHF) that meets New York Heart Association (NYHA) Class II to IV definitions (see Appendix 2) and/or ejection fraction \<40% by MUGA scan or \<50% by echocardiogram and/or MRI * A known history of sustained ventricular tachycardia (VT), ventricular fibrillation (VF), Torsade de Pointes, or cardiac arrest unless currently addressed with an automatic implantable cardioverter defibrillator (AICD) * Hypertrophic cardiomegaly or restrictive cardiomyopathy from prior treatment or other causes * Uncontrolled hypertension, defined as blood pressure (BP) of ≥160/95; patients who have a history of hypertension controlled by medication must be on a stable dose (for at least one month) and meet all other inclusion criteria * Any cardiac arrhythmia requiring an anti-arrhythmic medication (excluding stable doses of beta-blockers) * Patients taking drugs leading to significant QT prolongation within the specified wash out period (See Appendix 3: Medications That May Cause QTc Prolongation). * Concomitant use of CYP3A4 inhibitors