-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. 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 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

             NOTE: Because no dosing or adverse event data are currently available on the use of
             duvelisib combination with romidepsin and CC-486 (5-azacitidine) in patients <18 years
             of age, children are excluded from this study, but may be eligible for future
             pediatric trials.

          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

             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

          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 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.