Inclusion Criteria:  Pathologically-confirmed mature T-cell lymphomas at the enrolling institution.  Permitted histologies include:  i) Stage ≥Ib CTCL, which has relapsed or progressed after at least two systemic therapies. In order to ensure balanced enrollment for patients with systemic T-cell lymphoma and CTCL, a maximum of 15 CTCL patients will be enrolled in expansion cohort.  ii) Systemic anaplastic large cell lymphoma that has relapsed after therapy containing brentuximab vedotin.  iii) T-cell prolymphocytic leukemia (treatment naïve permitted)  For the following histologies, patients are required to have received at least 1 prior therapy:  iv) T-cell large granular lymphocytic leukemia  v) Aggressive NK-cell leukemia  vi) Adult T-cell leukemia/lymphoma  vii) Extranodal NK/T- cell lymphoma, nasal type  viii) Enteropathy-associated T-cell lymphoma  ix) Monomorphic epitheliotropic intestinal t-cell lymphoma  x) Hepatosplenic T cell lymphoma  xi) Subcutaneous panniculitis-like T-cell lymphoma  xii) Primary cutaneous anaplastic large cell lymphoma  xiii) Primary cutaneous gamma/delta T-cell lymphoma  xiv) Primary cutaneous CD8-positive aggressive epidermotropic cytotoxic T-cell lymphoma  xv) Peripheral T-cell lymphoma, not otherwise specified  xvi) Angioimmunoblastic T cell lymphoma  xvii) Follicular T-cell lymphoma  xviii) Nodal peripheral T-cell lymphoma wih T follicular helper phenotype  b) Age ≥18 years at time of enrollment  c) Performance status, as assessed in the ECOG grading system, ≤2  d) Laboratory criteria.  Laboratory criteria  i) For dose escalation phase:  Absolute neutrophil count ≥1.0 K/mcL (Note: growth factor is allowed) Platelet count ≥80 K/μl or ≥50 K/μl if due to lymphoma Calculated creatinine clearance ≥60mL/min by Cockcroft-Gault Total bilirubin ≤1.5x upper limit of normal (ULN) or ≤3x ULN if documented hepatic involvement with lymphoma, or ≤5x ULN if history of Gilbert's syndrome; AST and ALT ≤ 3x ULN; or ≤ 5x ULN if due to lymphoma involvement  ii) For dose expansion phase:  Absolute neutrophil count ≥1.0 K/mcL or ≥0.5 K/mcL if due to lymphoma or ≥0.0 K/mcL if due to T-PLL or large granular lymphocytic leukemia (LGL) (Note: growth factor is allowed). Platelet count ≥80 K/μl or ≥50 K/μl if due to lymphoma Calculated creatinine clearance ≥60mL/min by Cockcroft-Gault Total bilirubin ≤1.5x upper limit of normal (ULN) or ≤3x ULN if documented hepatic involvement with lymphoma, or ≤5x ULN if history of Gilbert's syndrome; AST and ALT ≤ 3x ULN; or ≤ 5x ULN if due to lymphoma involvement  e) Measurable disease, defined by at least one of the following:  °Revised International Working Group Classification for systemic lymphoma19  °Atypical T lymphocytes quantifiable by flow cytometry or morphology in the peripheral blood or bone marrow  mSWAT (Modified Severity Weighted Assessment Tool) >0  f) Ability to swallow pills  g) Women of reproductive potential* must have a negative serum or urine β human chorionic gonadotropin (βhCG) pregnancy test within 14 days of initiating therapy. All women of reproductive potential and all sexually active male patients must agree to use adequate methods of birth control (e.g. latex condoms) throughout the study and for 3 months after the last dose of study drug.  °*A woman of reproductive potential is a sexually-mature woman 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).  The effects of duvelisib on conception, pregnancy, and lactation are unknown. Since duvelisib has not been evaluated in pregnant or nursing women, the treatment of pregnant women or women of childbearing potential who are not using a highly effective contraception is contraindicated.  Exclusion Criteria:  Any serious medical condition, laboratory abnormality, or psychiatric illness that would prevent the subject from signing the informed consent form. Pregnant women. (Lactating women must agree not to breast feed while taking study medications). Prior allogeneic stem cell transplant. Prior use of duvelisib or ruxolitinib if either agent was discontinued due to toxicity. Previous systemic anti-cancer therapy for TCL within 14 days of initiating study drug a. 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.  b. Systemic corticosteroids must be tapered to 20mg/day or less prednisone (or equivalent) upon start of investigational treatment.  c. Topical steroids for CTCL is permitted on study.  f) Ongoing use of immunosuppressant medications, including corticosteroids greater than 20mg of prednisone or equivalent at the time of enrollment  g) History of chronic liver disease, veno-occlusive disease, or current alcohol abuse  h) Administration of a live vaccine within 6 weeks of first dose of study drug.  i) Prior surgery or gastrointestinal condition that may adversely affect drug absorption (e.g., gastric bypass surgery, gastrectomy)  j) Patients with HIV infection if they meet either of the below criteria:  i. detectable viral load ii. undetectable viral load with CD4 count <200 or not taking anti-retroviral medications.  k) Patients with chronic hepatitis B or C as defined by positive hepatitis B or C 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 must receive hepatitis B prophylaxis until at least 6 months after completion of study drug(s).  l) Subjects with active CMV (defined as positive CMV PCR with clinical manifestations consistent with active CMV infection) and requiring therapy. Carriers will be monitored per institutional guidelines.  m) Unable or unwilling to receive prophylaxis against pneumocystis, herpes simplex virus, or herpes zoster  g) Use of medications or consumption of foods that are strong inducers or inhibitors of CYP3A  Such agents must be discontinued at least 2 weeks prior to study intervention  Patients who (after enrollment) require use of a strong CYP3A4 inhibitor to treat a fungal/mold infection will require dose reductions n) Receipt of treatment for tuberculosis within 2 years prior to enrollment  o) Receiving 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. Early-stage cutaneous basal cell and squamous cell carcinomas are permissible  Adjuvant or maintenance therapy to reduce the risk of recurrence of other malignancy is potentially permissible after discussion with the MSK Principal Investigator.  p) Known central nervous system or meningeal involvement by TCL (in the absence of symptoms, investigation into central nervous system involvement is not required).  q) Unstable or severe uncontrolled medical condition (e.g., unstable cardiac function, unstable pulmonary condition) or any important medical illness that would, in the Investigator's judgment, increase the risk to the patient associated with his or her participation in the study.