Brief Summary: Hematopoietic progenitor cell (HPC- primitive cells in the blood, bone marrow and umbilical cord that can restore the bone marrow) transplant can be a curative therapy for the treatment of hematologic malignancies (a disease of the bone marrow and lymph nodes). The source of cells used for the transplant comes from related (sibling) and in cases where there is no sibling match, from unrelated donors through the National Marrow Donor Program. The availability of a suitable donor can be a significant obstacle for patients who need a transplant but do not have a matched donor. Cord blood that has been harvested from an umbilical cord shortly after birth has a rich supply of cells needed for transplant. These stored cord bloods are now being used to transplant adults without a matched donor Advantages to using cord blood includes a readily available source of cells with no risk to the donor during the collection process, immediate source of cells in urgent situations (no lengthy donor work-up)and a reduction in infectious disease transmission to the recipient. One of the main disadvantages is the cord blood has a small number of cells needed for transplant. In an adult, usually two cords are needed and large recipients do not qualify because they need too many cells. This study will use two different preparative regimens (chemotherapy and radiation) followed by one or two umbilical cord units (UBC). The preparative regimen used will be chosen by the physician and is based on patient's age, disease and medical condition at the time of transplant. Multiple objectives for this study include disease-free and overall survival, treatment related mortality, rate of cells taking hold, and the incidence and severity of the transplant complication called graft versus host disease (GVHD).

Detailed Description: Allogeneic hematopoietic cell transplantation (allo- HCT) is a curative therapy for the treatment of hematological and non-hematological malignancies and certain non-malignant conditions. Bone marrow or peripheral blood from a Human Leukocyte Antigen (HLA) matched sibling donor is the most commonly used source of allogeneic stem cells. However, HLA matched siblings are available for less than one third of the patients who require allo-Stem Cell Transplant (allo-SCT). In the absence of an HLA matched sibling, volunteer unrelated donors or partially mismatched related donors (PMRD), stored cord blood may be used as a source of allogeneic stem cells. Stored cord blood has been used as a source of allogeneic stem cells in infants and children, but had early skepticism in adults because of concerns about the engraftment potential of the relatively limited number of stem cells. The number of stem cells in a unit of cord blood is generally one log less than the number of stem cells on an average collection of bone marrow from an adult for transplantation. After the success of the first allogeneic umbilical cord blood transplantation in 1988, programs for banking screened unrelated donor CBSC have been initiated both in the United States and Europe. Dr Pablo Rubenstein started the first such bank at the New York Blood Center (NYBC) in 1993. Since its inception, the NYBC has provided unrelated donor cord blood stem cells for over 1000 transplants. Analysis of outcomes for the initial 562 transplant recipients from the NYBC revealed a cumulative rate of engraftment of 81% by day 42 for PMNs. and 85% by day 180 for platelets. Currently, approximately more than 100,000 cord blood units are available in cord blood banks worldwide and more than 2000 patients have received cord blood transplants from these banks. NetCord, an international cooperative group of cord blood banks, has developed a detailed set of standards for cord blood banking to facilitate international exchanges and to guarantee the quality of these products. Cord Blood Unit Selection: UCB units will be required to be a 4 to 6 of 6 HLA-A, -B antigen and -DRB1 allele match with the patient. Typing at HLA-C and -DQ will be obtained but not required in the match strategy. A minimum total nucleated cell (TNC) dose of \>2.0 x 107/kg at the time of freezing will be utilized when possible. When using double units, each unit should contain a minimum pre-cryopreserved TNC dose of 1.5 x 107/kg. UCB Transplant Procedure: There will be a myeloablative and reduced-intensity preparative regimen that can be given prior to infusion of cord product. The myeloablative approach will be selected in younger patients (\<50yo) with a HCT-CI score \<3. The reduced-intensity regimen will be selected for all older patients (\>50) or younger patients with a hematopoietic cell transplantation-specific comorbidity index (HCT-CI) score \>3. The reduced-intensity regimen will also be chosen for any patients being transplanted for indolent/follicular lymphomas, CLL, myeloma, or Hodgkin lymphoma; irrelevant of age or HCT-CI score. On a case by case basis, patients may receive a preparative regimen outside of their designated category as noted above with the approval of the PI, if deemed in the patient's best interest.