Doxorubicin, CC-(486) (5-azacitidine), Romidepsin, and Duvelisib (hARD) for T-cell Lymphoma

-INCLUSION CRITERIA: 1. Patients with newly diagnosed (Cohort 2) or relapsed/refractory (Cohorts 1 and 3) T-cell lymphoma (TCL) defined as follows (per 2016 WHO classification): * Adult T-cell leukemia/lymphoma * Extranodal NK-/T-cell lymphoma, nasal type * Enteropathy-associated T-cell lymphoma * Monomorphic epiteliotrophic intestinal T-cell lymphoma * Hepatosplenic T-cell lymphoma * Subcutaneous panniculitis-like T-cell lymphoma * Peripheral T-cell lymphoma, NOS * Angioimmunoblastic T-cell lymphoma * Follicular T-cell lymphoma * Nodal peripheral T-cell lymphoma with TFH phenotype * Anaplastic large-cell lymphoma, ALK+ and ALK- (only if relapsed/refractory after at least one line of systemic therapy, which must include brentuximab vedotin) * Breast implant-associated anaplastic large-cell lymphoma Note: For relapsed patients, prior treatment may include allogeneic stem cell transplantation 2. PTCL from initial diagnosis or recurrence must be histologically or cytologically proven and diagnosis be confirmed by the Laboratory of Pathology, NCI. 3. For patients without circulating lymphoma cells detectable by flow cytometry, a formalin fixed tissue block or 15 slides of tumor sample (archival or fresh) must be available at enrollment for performance of correlative studies. NOTE: Patients without circulating malignant cells must be willing to have a tumor biopsy if prior tissue or adequate archival tissue is not available (i.e., post-enrollment and prior to treatment). 4. Disease must be measurable with at least one measurable lesion by RECIL 2017 criteria or with an abnormal clonal T-cell population detectable by peripheral blood flow cytometry. 5. Age \>=18 years 6. ECOG performance status \<=2. 7. LVEF within institutional parameters as determined by echocardiography. 8. DLCO/VA Adjusted and FEV1 \>=50% of reference. 9. Adequate organ and marrow function as defined below: Absolute neutrophil count \>= 1,000/mcL Platelets \>= 75,000/mcL Hemoglobin \>= 9 g/dL (regardless of transfusion status) Total bilirubin \<= 1.5 X institutional upper limit of normal (ULN) AST(SGOT)/ALT(SGPT) \<= 2.5 X institutional ULN Creatinine clearance \>= 60 mL/min/1.73 m\^2 calculated by calculated using eGRF in the clinical lab 10. For women of childbearing potential (WCBP): negative serum \<= human chorionic gonadotropin ( \<=hCG) pregnancy test during screening. A negative pregnancy test is also required within 7 days before first treatment; screening results may be used for treatment if they fall within the required window. NOTE: WCBP is defined as sexually mature woman who has not undergone surgical sterilization or who has not been naturally postmenopausal for at least 12 consecutive months for women \>55 years of age) 11. Male and female participants of reproductive potential (i.e., not surgically sterile or female subjects who are not postmenopausal), must be willing to use a highly effective method of contraception for the duration of the study treatment and 3 months after the last dose of duvelisib and/or romidepsin and for 6 months after the last dose of doxorubicin, whichever is longer. 12. For patients that have received allogeneic HSCT, a minimum of 100 days must have elapsed before enrollment. Patients must have been off of immunosuppressive medications for prophylaxis of GVHD for at least four weeks before enrollment. 13. Ability of subject to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: 1. Patients who are receiving any other investigational agents. 2. Patients in need of immediate cytoreduction. 3. Chemotherapy or monoclonal antibody therapy within 4 weeks prior to start of treatment (6 weeks for nitrosoureas or mitomycin C); small molecule or radiation therapy within 2 weeks. 4. Prior lifetime doxorubicin therapy \>= 400 mg/m\^2. 5. Prior radiation therapy to chest with fields involving the heart. 6. Major surgery within 4 weeks prior to start of treatment 7. History of tuberculosis treatment within the 2 years prior to start of treatment 8. Administration of a live or live attenuated vaccine within 6 weeks prior to start of treatment, or of mRNA and adenovirus-based vaccines within 2 weeks prior to start of treatment 9. Patients who have received all of the planned study drugs - i.e., duvelisib, romidepsin, and CC-486 (5-azacitidine) - at any time point during prior treatments for TCL. Patients who have received one or two of the three drugs (alone or in combination) remain eligible. 10. Patients with previous malignant disease other than the target malignancy within the last 5 years with the exception of basal or squamous cell carcinoma of the skin or cervical carcinoma in situ. 11. Systemic treatment for acute or chronic graft versus host disease (GVHD) within 12 weeks of the first dose of duvelisib 12. Cohort 1 (Dose Escalation) only: history of grade 3/4 GVHD 13. Patients with active acute or chronic GVHD 14. History or concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function. Pulmonary function testing (PFTs) will be evaluated at screening. 15. Prior history of drug-induced colitis or drug-induced pneumonitis. 16. History of chronic liver disease or veno-occlusive disease 17. History of allergic reactions or known or suspected hypersensitivity attributed to compounds of similar chemical or biologic composition to duvelisib, romidepsin or mannitol and CC-486 (5-azacitidine). 18. Ongoing treatment with systemic steroids at a dose higher than 20 mg of prednisone (or equivalent) once daily 19. Patients with uncontrolled intercurrent illness including, but not limited to, detectable CMV or EBV viral load, and ongoing or active bacterial, fungal, or viral infection requiring systemic therapy, with the following exceptions: * Human immunodeficiency virus (HIV)-infected patients on effective anti- retroviral therapy with undetectable viral load within 6 months are eligible for this trial. * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable and on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Subjects on antimicrobial, antifungal, or antiviral prophylaxis. 20. History of inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis), celiac disease (i.e., sprue), prior gastrectomy or upper bowel removal, or any other gastrointestinal disorder or defect that would interfere with the absorption, distribution, metabolism or excretion of the study drugs and/or predispose the subject to an increased risk of gastrointestinal toxicity. 21. Clinically significant (i.e., active) cardiovascular disease: cerebral vascular accident/stroke (\< 6 months prior to enrollment), myocardial infarction (\< 6 months prior to enrollment), unstable angina, congestive heart failure (\>= New York Heart Association Classification Class II), congenital long QT syndrome, or other serious cardiac arrhythmia including 2nd degree atrio-ventricular (AV) block type II, 3rd degree AV block, or bradycardia (ventricular rate less than 50 beats/min). 22. Hypertrophic cardiomegaly or restrictive cardiomyopathy from prior treatment or other causes. 23. Uncontrolled hypertension, i.e., blood pressure (BP) of \>=160/95 at screening; 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. 24. Triplicate average baseline QTcF interval \>= 480 ms during screening 25. Patients taking drugs leading to significant QT prolongation Note: A 5 half-life washout period must have elapsed following the use of these drugs prior to administration of romidepsin. 26. Concomitant use of rifampin and other strong CYP3A4 inhibitors and inducers within 2 weeks prior to start of treatment. 27. Inability to receive prophylactic treatment for pneumocystis, herpes simplex virus (HSV), or herpes zoster (VZV). 28. Other severe acute or chronic medical conditions including psychiatric conditions such as recent (within the past year) or active suicidal ideation or behavior; or laboratory abnormalities that may increase the risk associated with study participation or study treatment administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the patient inappropriate for entry into this study 29. Pregnant or lactating women. NOTE: Pregnant women are excluded from this study because based on findings in animals and its mechanism of action, duvelisib can cause fetal harm when administered to a pregnant woman. In animal reproduction studies, administration of duvelisib to pregnant rats and rabbits during organogenesis caused adverse developmental outcomes including embryo-fetal mortality. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with duvelisib, breastfeeding should be discontinued if the mother is treated with duvelisib. These potential risks may also apply to other agents used in this study.