Azacitidine and Abatacept in Relapsed or Refractory T-Cell Lymphoma

Background: T-cell lymphoma is a blood cancer that affects immune system cells. People tend to survive less than 1 year if this disease does not respond to treatment (is refractory) or comes back after treatment (relapses). Azacitidine and abatacept are 2 drugs that are used to treat other diseases. Researchers want to know if these drugs, used together, can help people with T-cell lymphoma. Objective: To learn if azacitidine combined with abatacept can shrink tumors in people with T-cell lymphoma. Eligibility: People aged 18 years and older with T-cell lymphoma that either came back or did not respond to treatment. Design: Participants will be screened. They will have a physical exam with blood tests. They will have a test of their heart function. They will have imaging scans of their tumors. A sample of tumor tissue may be taken. Azacitidine is injected under the skin of the thigh, abdomen, or upper arm. Abatacept is infused through a needle inserted into a vein in the arm. Participants will receive the study drugs in 28-day cycles for up to 13 cycles. They will come to the clinic for each treatment. They will come to the clinic on day 1 and day 15 of the first cycle. After that, they will come to the clinic on the first 5 or 7 days of each cycle. Each clinic visit will take no more than 8 hours. Imaging scans and other tests will be repeated during the study. Participants will have follow-up visits for up to 5 years after they stop taking the study drugs....

-INCLUSION CRITERIA 1. Participants must have a histologically or cytologically confirmed T-cell lymphoma confirmed by the Laboratory of Pathology (LP), NCI. The one of the following T-cell lymphomas are included: * Peripheral T-cell lymphoma not otherwise specified (PTCL, NOS) * Angioimmunoblastic T-cell lymphoma (AITL) * T-follicular helper (TFH) lymphoma * Anaplastic large cell lymphoma (ALCL) * Enteropathy-associated T-cell lymphoma (EATL) * Monomorphic epitheliotropic intestinal T-cell lymphoma (MEITL) * Adult T-cell leukemia/lymphoma (ATLL) * Other T-cell lymphoma (TCL) 2. Participants must have a disease that is relapsed or refractory after initial systemic treatment. 3. Participants must have evaluable disease on screening imaging or by laboratory assessment. 4. Age \>= 18 years. 5. ECOG performance status \= 1,000 cells/mcL OR \>= 500 cells/mcL if bone marrow involvement with lymphoma * Platelets / \>= 50,000 cells/mcL OR \>= 25,000 cells/mcL if bone marrow involvement with lymphoma * Hemoglobin / \>= 8 g/dL (transfusions permitted) * Renal function / Serum creatinine \= 40 mL/min/1.73 m\^2 as estimated by the Modification of Diet in Renal Disease (MDRD). Note: there is no limit if involved by lymphoma. If not on target, a 24-hour urine creatinine clearance can be used to directly measure creatinine clearance. * Total bilirubin / \