A Phase I Trial Anti-CC Chemokine Receptor 4 Chimeric Antigen Receptor T Cells (CCR4 CAR T Cells) for CCR4 Expressing T-cell Malignancies Including Peripheral T-cell Non-Hodgkin Lymphoma (PTCL) and Cutaneous T-cell Non-Hodgkin Lymphoma (CTCL)

Background: * Chemokine receptor 4 (CCR4) the receptor for C-C chemokines CCL17 \[Thymus and activation-regulated chemokine (TARC)\] and CCL22 \[Macrophage-derived chemokine (MDC)\] is expressed on some subsets of normal human T lymphocytes including T regulatory cells (Tregs), TH2 and TH17 cells. It is also widely expressed in mature T-cell malignancies. * The anti-CCR4 antibody, mogamulizumab, has shown appreciable clinical activity against CCR4 expressing T-cell malignancies suggesting this molecule to be a robust therapeutic target. * A lentivirus vector-based 2nd generation anti-CCR4 chimeric antigen receptor (CCR4-CAR.4-1BB-CD3Zeta) has been generated, and its functionality and specificity have been validated in in-vitro cell lysis assays with several patient-derived CCR4+ malignant T cells lines as has its efficacy in a murine xenograft preclinical model of ATL. Objectives: -Determine safety of administering autologous CCR4 CAR T cells. Eligibility: * Diagnosis of relapsed/refractory mature T cell malignancy that expresses CCR4 including: Peripheral T-cell Lymphoma not otherwise specified (PTCL-NOS), angioimmunoblastic T-cell lymphoma (AITL), anaplastic large cell lymphoma (ALCL), hepatosplenic T-cell lymphoma (HSTCL), monomorphic epithelialtropic intestinal lymphoma (MEITL), enteropathy associated T-cell lymphoma (EATL) or cutaneous Tcell lymphoma (CTCL) including mycosis fungoides and subacute panniculitis-like Tcell Lymphoma or lymphomatous subtypes of ATL without evidence of CNS involvement or substantial circulating disease. * Age \>=18 years. * Adequate organ function and absence of medical conditions that would preclude safe administration of this treatment. Design: * Single-institution, open-label, non-randomized, 3+3 pilot dose escalation trial * Participants will undergo leukapheresis to obtain T-cells that will be genetically modified to express anti-CCR4 CARs on T-cells. Groups of 3 to 6 participants will receive autologous CCR4 CAR T cells at doses of 0.3 x 10\^6/kg, 1 x 10\^6/kg and 3 x 10\^6/kg provided that DLT has not been reached; If the first dose level exceeds the MTD, a subsequent cohort of 3-6 participants will be treated at the dose level of 0.1 x 10\^6/kg. * Up to 9 additional participants will be treated at the MTD or the maximum administered dose to better assess the activity of this treatment regimen. * Participants will be monitored for toxicity, antitumor effects and persistence of CCR4 CAR T-cells and assessed for response with periodic re-staging imaging and laboratory testing. * Statistical analyses will be limited to summary statistics by dose level and for all participants for safety parameters, biologic effects, response rate and survival.