Relapsed Malignant Blood Cancer After Allogeneic Hematopoietic Stem Cell Transplantation

* INCLUSION CRITERIA: RECIPIENT SUBJECTS: 1. Individuals who are candidates for allotransplant therapy for hematologic malignancies and are being evaluated at the Clinical Center for planned allotransplantation. 2. Individuals who have received allotransplant treatment for hematologic malignancy and have: 1. Hematologic recovery after allotransplant: e.g., have had neutrophil recovery to 500 cells/mcL. Secondary cytopenias or cytopenias due to disease progression will be permitted. Note: this requirement will not apply to subjects enrolling pre-transplant, i.e, who receive transplant-related medical care at the Clinical Center (CC). 2. An ongoing relationship with a primary oncologist who will continue to provide continuity of care during and after study participation. 3. Following record review and information exchange between the patients primary oncologist and the National Cancer Institute (NCI) Principal Investigator (PI)/Designee, the PI/Designee determines that the individual reasonably could be expected to safely tolerate travel to and from the Clinical Center (CC) to undergo evaluation as defined in the protocol, in the event that the patient is ineligible or uninterested in participating in open treatment protocols. 3. 18-99 years. 4. Ability of subject to understand and the willingness to sign a written informed consent document. DONOR SUBJECTS: 1. Individuals who are/will be the donors of allogeneic hematopoietic stem cell transplants received by Recipient-Subjects who are to be enrolled on this protocol. 2. Age 18-99 years. 3. Ability of the subject to understand and the willingness to sign a written informed consent document. 4. Individuals with evidence of infection with transfusion-transmittable agents (Hepatitis B and C Viruses (HBV, HCV); Human Immunodeficiency Virus (HIV (Omega)), Human TLymphotrophic Virus (HTLV I/II), West Nile Virus (WNV) and Trypanosoma cruzi) will not be excluded from study participation. However, Donor-Subjects with evidence of HIV infection will only be able to donate cells for research. Donors with a history of HBV or HCV infection will be able to donate for research, and may be eligible to donate for therapeutic administration. However, determination of permissibility for clinical donation will require a hepatology consultation and the consent of the intended recipient after discussion of the risk/benefit of the donor cell product and the possibility/consequences of transmission. The PI/Designee will make the final determination of permissibility of donation for recipient cell therapy. 6\. Unrelated donor selection will be in accordance with the National Marrow Donor Program (NMDP) standards. When a potentially eligible recipient of an unrelated donor product from an NMDP Center is identified, the recipient will complete an NMDP search transfer request to allow NIH NMDP staff to contact the NMDP Coordinating Center, who will, in turn, contact the donors prior Donor Center. The NMDP Policy for Subsequent Donation Requests will be followed and the appropriate forms (Subsequent Donation Request Form and Therapeutic T Cell Collection Prescription Form) will be submitted as required. EXCLUSION CRITERIA: RECIPIENT SUBJECTS: 1. Individuals with rapid disease progression or aggressive cancer histology who, in the opinion of the PI/Designee, require urgent therapy within 30 days in order to preserve organ function or quality of life. This restriction will not apply if there is no approved therapy with a reasonable chance of disease response, if the patient does not have access to an effective therapy and the patient appears to be eligible for an accruing CC treatment protocol or if the patient is enrolled on an NIH/CC clinical protocol, e.g., allotransplant protocol. 2. Pregnancy or lactating. Additionally, Recipient-Subjects of childbearing potential that will receive cancer treatment under this protocol must be willing to use an effective method of contraception. DONOR SUBJECTS: 1\. Adult donors who are not eligible for clinical donation will not be excluded from study participation, but will only be able to donate cells for research.