Matched sibling, cord blood and haploidentical donors are not eligible. Additional Patient Inclusion Criteria for Patients Assigned to Double Umbilical Cord Blood Arm: DONOR: HLA-matched sibling cord blood exclusions: Any cord blood units without full maternal testing and negative results for hepatitis A, B, C, HIV, and HTLV- viruses; any additional available virology results on the unit itself will be reviewed but are not mandated, complete or always available; cord blood units are presumed to be cytomegalovirus (CMV) negative regardless of serologic testing due to passive transmission of maternal CMV antibodies DONOR: Unrelated Umbilical Cord Blood: The patient and the cord blood unit(s) must be matched for at least of loci as defined above DONOR: Unrelated Umbilical Cord Blood: Selection of two umbilical cord blood (UCB) units is allowed to provide sufficient cell dose DONOR: Unrelated Umbilical Cord Blood: The UCB unit with the least HLA disparity (with the patient) will be selected first (i.e., selection priority is / match > / match> / match); additional UCB units then may be selected to achieve the required cell dose; if a second unit is required, this unit will be the unit that most closely HLA matches the patient and meets minimum size criteria outlined below of at least . x ^ TNC/kg (ie a smaller more closely matched unit will be selected over a larger less well matched unit as long as minimum criteria are met) DONOR: Unrelated Umbilical Cord Blood: Each UCB unit MUST contain at least . x ^ TNC per kilogram recipient weight DONOR: Unrelated Umbilical Cord Blood: The total cell dose of the combined units must be at least . x ^ TNC per kilogram recipient weight DONOR: Unrelated Umbilical Cord Blood: Any cord blood units with < . x ^ total nucleated cells per kilogram recipient weight DONOR: Unrelated Umbilical Cord Blood: Any cord blood units without full maternal testing and negative results for hepatitis A, B, C, HIV, and HTLV- viruses; any additional available virology results on the unit itself will be reviewed but are not mandated, complete or always available; cord blood units are presumed to be CMV negative regardless of serologic testing due to passive transmission of maternal CMV antibodies Cord blood product manufactured by the NCBP (at least one, if the graft contains more than one units) Patient with either one or both:\r\n* Two / HLA or better high resolution matched umbilical cord blood (UCB) grafts with a cell dose of . x ^ total nucleated cell (TNC)/kg each, or\r\n* A related haplo-identical donor Patients must have two cord blood (CB) units available which are matched with the patient at , , or / HLA\r\nclass I (serological) and II (molecular) antigens. Each cord must contain at least . x ^ total nucleated cells/Kg recipient body weight (pre-thaw). Patients will have a back-up graft from any of the following: an available fraction of autologous marrow; or peripheral blood progenitor cells (PBPCs) harvested and cryopreserved; or family member donor; or a third cord blood unit. Availability of at least one / HLA-matched (HLA-A, B, and DR loci) cord blood unit from the National Marrow Donor Program (NMDP). The cord blood unit must contain a minimum total nucleated cell (TNC) (prior to thawing) of at least x^ cells per kilogram of recipient body weight Patients with any hematologic malignancy undergoing either an unmodified allogeneic HCT or a double umbilical cord blood transplant with or without the infusion of T-cell depleted HLA-haploidentical peripheral blood stem cells UCB graft selected according to current University of Minnesota umbilical cord blood graft selection algorithm DONOR: Peripheral blood is preferred over bone marrow for non-umbilical cord blood recipients DONOR: Umbilical cord blood units will be selected according to the following umbilical cord blood graft selection criteria; one or cord blood (CB) units may be used to achieve the required cell dose DONOR: Selection of two CB units is allowed to provide sufficient cell dose (see below for algorithm to determine single versus double unit transplant); when multiple units are selected, the following rules apply:\r\n* The CB unit with the least HLA disparity (with the patient) will be selected first (i.e., selection priority is / match > / match > / match); additional CB units then may be selected to achieve the required cell dose, as outlined below; if a second unit is required, this unit will be the unit that most closely HLA matches the patient and meets minimum size criteria outlined below of at least . x ^ total nucleated cells (TNC)/kg (i.e. a smaller, more closely matched unit will be selected over a larger, less well matched unit as long as minimum criteria are met)\r\n* If two CB units are used:\r\n** The total cell dose of the combined units must be at least . x ^ TNC per kilogram recipient weight\r\n** Each CB unit MUST contain at least . x ^ TNC per kilogram recipient weight\r\n* Algorithm for determining single versus double unit cord blood transplant:\r\n** Match grade /: TNC dose >= . x ^/kg\r\n** Match grade /, /: TNC dose >= . (+/- .) x ^/kg DONOR: General comments:\r\n* Units will be selected first based on the TNC dose and HLA matching\r\n* Cluster of differentiation (CD)+ cell dose will not be used for unit selection unless units of equal HLA-match grade and similar TNC dose (+/- . x ^ TNC/kg) are available; in this case, the unit with the larger CD+ cell dose (if data available) should be selected\r\n* A CB unit that is / mismatched but homozygous at the locus of mismatch should be chosen over a / unit with bidirectional mismatch even if the latter unit is larger (has more cells); this also applies to / units; this is only applicable to choosing units within a given match grade\r\n* Other factors to be considered: \r\n** Within the same HLA match grade, matching at DR takes preference\r\n** Cord blood banks located in the United States are preferred\r\n* Up to % of the cord blood product(s), when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; these products will be used to conduct studies involving the kinetics of engraftment and immunobiology of double cord transplantation PART : Patients without an HLA-identical or -allele-mismatched related donor or unrelated donor or umbilical cord blood units that meet transplant criteria Subjects must have a back-up umbilical cord on the registry in addition to the umbilical cord being used in this study Patients must have histo-pathologically confirmed diagnosis of hematologic malignancy (leukemia or lymphoma); patients to be considered will either have resistant/refractory hematologic malignancies (disease exceeding % by morphology or with measurable extramedullary diseases: e.g. nodal disease or chloroma) or have relapsed following an initial allogeneic HSCT; alternatively, they will have a hematologic malignancy where allogeneic HSCT is indicated and their disease is in a state of remission, but they lack a well matched donor: related, unrelated donor (URD) or umbilical cord blood (UCB) HSC source (less than of URD match, less than of UCB match and/or less than x^ CD + HSC/kg in a matched UCB product); patients between and years of age will have been previously diagnosed and cared for by a Pediatric Oncologist DONOR: Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria in the table below\r\n* Match grade \r\n** /\r\n*** Single unit allowed for total nucleated cell (TNC) dose >= . x ^/kg\r\n** /, /\r\n*** Single unit allowed for TNC dose >= . (+/- .) x ^/kg\r\n* If two CB units are used, the total cell dose of the combined units must be at least . x ^ TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of . x ^ TNC/kg DONOR (FHCRC only): Up to % of the unmanipulated cord blood product (s), when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; these products will be used to conduct studies involving the kinetics of engraftment and immunobiology of double cord transplantation DONOR: Any cord blood units with < . x ^ total nucleated cells per kilogram recipient weight DONOR: Any cord blood units without full maternal testing and negative results for hepatitis A, B, C, HIV, and human T-lymphotropic virus (HTLV)- viruses; any additional available virology results on the unit itself will be reviewed but are not mandated, complete or always available; cord blood units are presumed to be CMV negative regardless of serologic testing due to passive transmission of maternal CMV antibodies DONOR: Any cord blood units with < . x ^ total nucleated cells per kilogram recipient weight DONOR: Any cord blood units without the full maternal testing and negative results for hepatitis A, B, C, HIV, and human T-lymphotropic virus (HTLV-) viruses; any additional available virology results on the unit itself will be reviewed but are not mandated, complete or always available; cord blood units are presumed to be cytomegalovirus (CMV) negative regardless of serologic testing due to passive transmission of maternal CMV antibodies DONOR: HPC-cord blood (CB) units will be obtained from established cord blood banks including, but not limited to: the National Marrow Donor Program, New York National Cord Blood Program, St. Louis Cardinal Glennon Cord Blood Bank, and University of Colorado Cord Blood Bank; umbilical cord blood unit (UCB) grafts will meet criteria established in Childrens Memorial Hospital Stem Cell Transplant Program policy VII-B entitled Allogeneic Donor Identification, Evaluation, Education, Consent and Management Umbilical cord blood used as an unrelated stem cell source will provide > . x ^ cells/kg and will be matched at - of HLA A, B, and DRBI loci; cord blood grafts may include a single or pair of cord units depending on the cell dose Highly-matched unrelated umbilical cord blood (UCB) (> / matches identified by high-resolution typing) accepted if a sibling donor is not able to donate bone marrow AND UCB with a sufficient number of nucleated cells (NCs) (i.e., > . x ^/kg recipient body weight [BW]) is cryopreserved Suitable UCB units available according to Umbilical Cord Blood Graft selection algorithm; the UCB graft may consist of one or two UCB units UCB units will be selected according to current University of Minnesota umbilical cord blood graft selection algorithm; one or UCB units may be used to achieve the required cell dose Patients must have two CB units available which are matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens; each cord must contain at least . x ^ total nucleated cells/Kg recipient body weight (pre-thaw); cord blood units will be procured through the National Marrow Donor Program (NMDP) PBSCs from identical twins are permitted, but no other allogeneic cells are allowed; sse of autologous stored umbilical cord blood stem cells is not allowed Patient must have hematologic malignancy that meets institutional eligibility requirements for cord blood transplant Receiving umbilical cord blood Cord blood transplant Acute GVHD developing after allogeneic hematopoietic cell transplantation using either bone marrow, peripheral blood, or umbilical cord blood. Recipients of non-myeloablative, reduced intensity conditioning and myeloablative transplants are eligible. All allogeneic donor sources are permitted, including siblings, unrelated donors, human leukocyte antigen (HLA)-haploidentical related donors and umbilical cord blood. A back-up graft identified, in case of graft failure, from any of the following sources: an available fraction of autologous marrow; or peripheral blood progenitor cells (PBPCs) harvested and cryopreserved; or family member donor; or a third cord blood unit Patients must have two cord blood units available which are matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens; each cord must contain at least million total nucleated cells/Kg recipient body weight (pre-thaw) Prior recipient of cord blood Availability of suitable primary and secondary umbilical cord blood (UCB) units. Has an available HCT donor or identified cord blood unit. Related and unrelated donors, and bone marrow, peripheral blood, or cord blood stem cell sources allowed Patients must have two cord blood (CB) units available which are matched with the patient at , , or / human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens; each cord must contain at least . x ^ total nucleated cells/kg recipient body weight (pre-thaw) Umbilical cord HCT Any patient who has undergone a single or double umbilical cord blood transplant (UCBT) except those with primary myelofibrosis All graft sources will be eligible (bone marrow, peripheral blood or umbilical cord blood) Three umbilical cord blood (UCB) units composing the graft will be selected according to the current University of Minnesota umbilical cord blood graft selection algorithm Allogeneic stem cell transplantation using a single or multiple umbilical cord blood units or using bone marrow Have received first peripheral blood, marrow or cord blood transplant from a family or unrelated donor for hematologic malignancy or myeloproliferative disorder UCB unit(s) composing the graft will be selected according to the current University of Minnesota umbilical cord blood graft selection algorithm plus an additional cord blood unit to be used as the source to manufacture the Treg product; this UCB unit must be matched at -/ to the patient, considering HLA-A, B at the antigen level and DRB at the allele level Unrelated donor, umbilical cord blood, mismatched, or haploidentical transplants