Inclusion Criteria:

          -  Pathologically confirmed T-cell lymphomas at the enrolling institution, including
             stage ≥ Ib CTCL, which has relapsed or progressed after at least one systemic therapy.

          -  Age ≥ 18

          -  Previous systemic anti-cancer therapy must have been discontinued at least 3 weeks
             prior to treatment. For the dose expansion phase, in progressing subjects, a 2 week
             washout may be allowed after discussion with the MSK Principal Investigator.

          -  Previous radiation and/or surgery must have been discontinued or completed at least 2
             weeks prior to treatment in this study and adverse effects must have resolved to Grade
             1 or baseline. Lymph node or other diagnostic biopsies within 2 weeks are not
             considered exclusionary.

             ° Patients who have received localized RT as part of their immediate prior therapy may
             be allowed to enroll with shorter washout period after discussion with the MSK
             Principal Investigator.

          -  ECOG ≤ 2

          -  Meet the following laboratory criteria without use of growth factor support or
             platelet transfusions for 1 week:

             i) Absolute neutrophil count ≥ 1.0 K/mcl, ii) Platelet count ≥ 80 K/μl (in the
             expansion cohorts, if thrombocytopenia is due to bone marrow involvement platelet
             count must be ≥ 50 K/μL), iii) Patients enrolled in the dose escalation phase who are
             not enrolled on the expansion cohorts must have calculated creatinine clearance ≥
             50ml/min by Cockcroft-Gault formula. Patients enrolled in the Dose Expansion phase
             must have calculated creatinine clearance ≥ 40ml/min by Cockcroft-Gault formula.

        iv) Total bilirubin ≤ 1.5 x upper limit of normal (ULN) or ≤ 3 x ULN if documented hepatic
        involvement with lymphoma, or ≤ 5 x ULN if history of Gilbert's syndrome; AST (SGOT) and
        ALT (SGPT) ≤ 3 x ULN; ≤ 5 ULN if due to lymphoma involvement

          -  Measurable disease for dose expansion and lead in phase only.

        Measurable disease defined by:

          1. Revised International Working Group (Cheson, 2007) Classification for systemic
             lymphoma or

          2. Atypical and or malignant lymphocytes quantifiable by flow cytometry or morphology in
             blood

          3. or bone marrow mSWAT > 0 or Sezary couny >/= 1000 cells/ul

               -  Short course systemic corticosteroids for disease control, improvement of
                  performance status or non-cancer indication (< 7 days) must have been
                  discontinued at least 6 days prior to study treatment. Stable ongoing
                  corticosteroid use (≥ 30 days) up to an equivalent dose of 20 mg of prednisone is
                  permissible.

                  i) Topical steroids that have been used for > 3 weeks may be continued (CTCL
                  only). All other histologies (not CTCL): Topical steroids use is permissible
                  without restriction

               -  Women of reproductive potential† must have a negative serum or urine β human
                  chorionic gonadotropin (βhCG) pregnancy test. All women of reproductive
                  potential, all sexually active male patients, and all partners of patients must
                  agree to use adequate methods of birth control (e.g. latex condoms) throughout
                  the study and for 30 days after the last dose of study drug.

                    -  A female of reproductive potential is a sexually mature female who: has not
                       undergone a hysterectomy or bilateral oophorectomy; or has not been
                       naturally postmenopausal for at least 24 consecutive months (i.e. has had
                       menses at any time in the preceding 24 consecutive months).

        Exclusion Criteria:

          -  Any serious medical condition, laboratory abnormality, or psychiatric illness that
             would prevent the subject from signing the informed consent form.

          -  Pregnant females. (Lactating females must agree not to breast feed while taking the
             study medications).

          -  Prior use of duvelisib if discontinued due to toxicity.

          -  For the romidepsin arm of the study, prior therapy with romidepsin if discontinued due
             to toxicity.

          -  For the bortezomib arm of the study, prior therapy with a proteasome inhibitor if
             discontinued due to toxicity.

          -  For the bortezomib arm of the study, patients with grade ≥2 peripheral neuropathy.

          -  History of chronic liver disease, veno-occlusive disease, or current alcohol abuse.

          -  Administration of a live vaccine within 6 weeks of first dose of study drug.

          -  Prior surgery or gastrointestinal dysfunction that may affect drug absorption (e.g.,
             gastric bypass surgery, gastrectomy)

          -  Known seropositive and requiring anti-viral therapy for human immunodeficiency virus
             (HIV), hepatitis B virus (HBV) or hepatitis C virus (HCV). Subjects with chronic
             hepatitis B or C as defined as test .

          -  Subjects with positive Hep B serology. Subjects with a negative HBsAg and a positive
             HBcAb require an undetectable/negative hepatitis B DNA test (e.g., polymerase chain
             reaction [PCR] test) to be enrolled, and will require prophylactic antiviral treatment
             (e.g., F) initiated prior to the first dose of study drug, an continued until
             approximately 6 to 12 months after completion of study drug(s).

          -  Patients with positive hepatitis C virus Ab

          -  Subjects with active EBV unrelated to underlying lymphoma (positive serology for
             anti-EBV VCA IgM antibody and negative for anti-EBV EBNA IgG antibody, or clinical
             manifestations and positive EBV PCR consistent with active EBV infection.

          -  Subject with active CMV (positive serology for anti-CMV IgM antibody and negative for
             anti-CMV IgG antibody and positive CMV PCR with clinical manifestations consistent
             with active CMV infection) and requiring therapy will be excluded from participation
             in the study. Carriers will be monitored per institutional guidelines.

          -  Receiving systemic therapy for another primary malignancy (other than T-cell lymphoma)

          -  Patients with more than one type of lymphoma may be enrolled after discussion with the
             MSK Principal Investigator.

          -  Adjuvant or maintenance therapy to reduce the risk of recurrence of other malignancy
             (other than T-call lymphoma) is permissible after discussion with the MSK Principal
             Investigator.

          -  Known central nervous system or meningeal involvement (in the absence of symptoms,
             investigation into central nervous system involvement is not required).

          -  Uncontrolled infection requiring systemic antimicrobials

          -  The following known cardiac abnormalities:

               1. Congenital long QT syndrome.

               2. QTc/QTf interval ≥ 480 milliseconds; unless secondary to pacemaker or bundle
                  branch block.

               3. Myocardial infarction within 6 months. (Subjects with a history of myocardial
                  infarction within the last 6 to12 months 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.)

               4. Other significant ECG abnormalities including 2nd degree atrio- ventricular (AV)
                  block (AV) block type II, 3rd degree AV block.

               5. Symptomatic coronary artery disease (CAD), e.g., angina Canadian Class II-IV (see
                  Appendix B). 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.

                  II-IV (see Appendix B). 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.

               6. 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.

               7. Congestive heart failure (CHF) that meets New York Heart Association (NYHA) Class
                  II to IV definitions (see Appendix C) and/or ejection fraction <45% by MUGA,
                  echocardiogram, or cardiac MRI.

               8. 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).

               9. Hypertrophic cardiomegaly or restrictive cardiomyopathy from prior treatment or
                  other causes.

              10. Uncontrolled hypertension, i.e., blood pressure (BP) of ≥170/95; patients who
                  have a history of hypertension controlled by medication must be on a stable dose
                  (for at least one month prior to study registration) and meet all other inclusion
                  criteria.

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

              12. For patients enrolling on the Romidepsin arm; taking drugs associated with
                  significant QTc/QTf prolongation, unless able to be switched to non-QTc/QTf
                  prolonging medication or on a stable dose without significant QT prolongation
                  (>470 msec). Caution should be used when administering study drugs to patients
                  taking medications significantly metabolized by these enzymes refer to
                  (http://medicine.iupui.edu/clinpharm/ddis/clinical-table/) for clinically
                  relevant medications. Particular attention should be paid to patients receiving
                  warfarin. Patient should have coagulation parameters monitored regularly, and
                  warfarin dose adjusted accordingly. If these drugs cannot be discontinued or
                  replaced enrollment may be allowed after discussion with MSK PI.