Inclusion Criteria (these criteria apply to both the phase 1 and phase 2 portion of the
        study)

          -  Age >18 years at the time of signing the informed consent

          -  Patients must have histologically confirmed newly diagnosed (ND) or
             Relapsed/Refractory Peripheral T-Cell Lymphoma (R/R PTCL) defined according to the
             2016 World Health Organization (WHO) classification criteria.

          -  Patients with R/R PTCL who have received at least one previous line of therapy are
             eligible to be enrolled in this study.

          -  Patients who are candidate for an autologous or allogeneic stem cell transplantation
             (SCT) will be allowed to receive the study drugs as a "bridge" to transplantation.

          -  Evaluable (phase 1) or measurable (phase 2) disease.

          -  Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2

          -  Patients must have adequate organ and marrow function as defined by:

               1. Aspartate transaminase (AST) and alanine transaminase (ALT) levels ≤ 2 x
                  institutional upper limit of normal (ULN); total bilirubin ≤ 1.5 x ULN (AST, ALT,
                  and total bilirubin ≤ 3 × ULN in subjects with documented Gilbert's syndrome or
                  hyperbilirubinemia clearly attributed to lymphoma involvement of the liver);

               2. Creatinine levels < 2 mg/dL; or creatinine clearance > 40 mL/min

               3. Absolute neutrophil count (ANC) > 1,000/μL; platelet count > 75,000/μL

               4. Negative urine or serum pregnancy test for women of childbearing potential. All
                  women of childbearing potential must agree to use an effective barrier method of
                  contraception (either an intrauterine device (IUD) or double barrier method using
                  condoms or a diaphragm plus spermicide) during the treatment period and for at
                  least 1 month after discontinuation of the study drugs. Male subjects should use
                  effective barrier method of contraception during the treatment period and for at
                  least 3 months after discontinuation of the study drugs.

          -  Ability to understand and the willingness to sign a written informed consent document.

          -  Ability to adhere to the study visit schedule and other protocol requirements

        Exclusion Criteria (these criteria apply to both the phase 1 and phase 2 portion of the
        study)

          -  Prior Therapy (for patients with R/R PTCL)

               1. Exposure to any agent targeting PD-1, PD-L1 or cytotoxic T-lymphocyte-associated
                  protein 4 (CTLA-4).

               2. For three-drug combinations, the patient must not have been exposed to at least
                  two of those drugs. For two drugs combinations patients must not have been
                  exposed to any of those drugs. Patients will be enrolled in the first treatment
                  Arm that satisfies this exclusion criterion, according to the following sequence:
                  Arm A; Arm B; Arm C; Arm D.

               3. Exposure to biologic therapy, immunotherapy, chemotherapy, investigational agent
                  for malignancy, or radiation therapy within 2 weeks prior to entering the study
                  or lack of resolution of AE due to previously administered antineoplastic therapy
                  to grade 1 or less according to National Cancer Institute (NCI) Common
                  Terminology Criteria for Adverse Events (CTCAE), Version 4.0.

               4. Current or prior use of immunosuppressive medication within 14 days prior to
                  first dose of durvalumab. The following are exceptions to this criterion:
                  intranasal, inhaled, topical or local steroid injections (e.g., intra-articular
                  injection); steroids as premedication for hypersensitivity reactions; systemic
                  corticosteroid at physiologic doses not to exceed 10 mg/day of prednisone or
                  equivalent for at least 5 days prior to the start of the study drugs.

               5. Prior allogeneic SCT

          -  History of, or suspected allergic reactions to, durvalumab, pralatrexate, oral
             5-azacitidine, or romidepsin or any of their excipients.

          -  For patients who are treated with oral 5-azacitidine, any gastrointestinal disorder
             that would interfere with the absorption of the study drug

          -  Concomitant use of CYP3A4 inhibitors

          -  Uncontrolled intercurrent illness.

          -  Any of the following cardiac abnormalities (only for patients receiving romidepsin):

               1. Congenital long QT syndrome;

               2. corrected QT (QTc) interval ≥ 501 milliseconds;

               3. Patients taking drugs leading to significant QT prolongation (See Appendix 3);

               4. Myocardial infarction within 6 months of cycle 1, day 1. [Subjects with a history
                  of myocardial infarction between 6 and 12 months prior to cycle 1, day 1, 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];

               5. 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);

               6. Symptomatic coronary artery disease (CAD), e.g., angina Canadian Class II-IV 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;

               7. 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;

               8. Congestive heart failure (CHF) that meets New York Heart Association (NYHA) Class
                  II to IV definitions (see Appendix 5) and/or ejection fraction < 40% by
                  multitargeted acquisition (MUGA) scan or < 50% by echocardiogram and/or magnetic
                  resonance imaging (MRI);

               9. 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);

              10. Hypertrophic cardiomegaly or restrictive cardiomyopathy from prior treatment or
                  other causes;

              11. Uncontrolled hypertension, i.e., 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; or

              12. Any cardiac arrhythmia requiring an anti-arrhythmic medication (excluding stable
                  doses of beta-blockers).

          -  Pregnancy or breast-feeding.

          -  Active concurrent malignancy (except non-melanoma skin cancer, prostatic
             intraepithelial neoplasia, or carcinoma in situ of the cervix). Patients whose
             lymphoma has transformed from a less aggressive histology remain eligible.

          -  Receipt of solid organ transplant.

          -  Active or prior documented autoimmune or inflammatory disorders (including
             inflammatory bowel disease [eg, colitis, Crohn's disease], diverticulitis with the
             exception of a prior episode that has resolved or diverticulosis, celiac disease,
             irritable bowel disease, or other serious gastrointestinal chronic conditions
             associated with diarrhea; systemic lupus erythematosus; Wegener's syndrome
             [granulomatosis with polyangiitis]; myasthenia gravis; Graves' disease; rheumatoid
             arthritis; hypophysitis, uveitis; etc) within the past 3 years prior to the start of
             treatment. The following are exceptions to this criterion: subjects with vitiligo or
             alopecia; subjects with hypothyroidism (eg, following Hashimoto syndrome) stable on
             hormone replacement; or subjects with psoriasis not requiring systemic treatment.

          -  Central nervous system (CNS) involvement, including lymphomatous meningitis.

          -  Known active hepatitis A, B or C virus infection.

          -  Known HIV infection.

          -  History of primary immunodeficiency.

          -  Receipt of live, attenuated vaccine within 30 days prior to study entry. Enrolled
             patients should not receive live vaccine during the study and 30 days after the last
             dose of durvalumab.