Patient must have a history of solid organ transplantation Patients who have received prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible No prior solid organ transplantation. Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Solid tumor transplantation. History of solid organ transplantation. Prior solid organ transplantation. Prior solid organ transplantation. Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients with prior solid organ transplantation on anti-immunosuppressant Has history of solid organ transplantation Patients who have received a prior solid organ transplantation are not eligible Patients who have previously received solid organ transplantation are NOT eligible for either Stratum History of solid organ transplantation Patients who have received a prior solid organ transplantation are not eligible Having received a solid organ transplantation They must not have a history of solid organ transplantation. Subjects who have received prior solid organ transplantation are not eligible Patients who have received a solid organ transplantation are not eligible. Prior solid organ transplantation Participant has received allogeneic HSCT or solid organ transplantation. Subjects who have received prior solid organ transplantation are not eligible Patients who have received prior solid organ transplantation are not eligible Prior solid organ transplantation Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible Patients who have received a prior solid organ transplantation are not eligible History of solid organ transplantation History of solid organ transplantation Prior solid organ transplantation History of primary immunodeficiency, solid organ transplantation, or tuberculosis Patients who have received a prior solid organ transplantation are not eligible Patients with a history of solid organ transplantation History of primary immunodeficiency, solid organ transplantation, or tuberculosis Patients who have received a prior solid organ transplantation are not eligible Subject has received allogeneic HSCT or solid organ transplantation. Solid organ transplantation requiring immunosuppression History of organ allograft Subjects with organ allograft requiring immunosuppression Patient with an organ allograft Prior HCT (allograft or prior autograft) No prior organ allograft Receipt of an organ allograft. No prior organ allograft or allogenic bone marrow transplantation History of liver allograft; prior hepatic resection is allowed No history of organ allograft History of organ allograft, stem cells or bone marrow transplant Prior organ allograft or allogenic bone marrow transplantation Presence of any tissue or organ allograft, regardless of need for immunosuppression, including corneal allograft; patients with a history of allogeneic hematopoietic stem cell transplant will also be excluded DONOR: Unable to provide a bone marrow allograft product Prior organ allograft Prior allograft or prior autograft Subjects required to take corticosteroids or other immunosuppressive therapy such as those with organ allograft Donor must provide a marrow allograft. However, abnormalities of specific organs will not be considered grounds for exclusion if they are the result of the EBV+ malignancy or its treatment (e.g. a renal allograft recipient with an EBVLPD may be on dialysis because the allograft was rejected when the immune suppression was stopped as a first approach to treatment of the EBVLPD); a the discretion of the investigator, patients with elevated but stable creatinine will not be precluded from treatment on study Prior organ allograft or allogeneic transplantation. History of organ allograft or other history of immunodeficiency Prior organ allograft or allogeneic transplantation Prior organ allograft or allogeneic transplantation, if the transplanted tissue is still in place Prior organ allograft or allogeneic transplantation, if the transplanted tissue is still in place Patients may have failed prior allograft Prior organ allograft or allogeneic transplantation, if the transplanted tissue is still in place No known prior organ allograft or allogeneic transplantation Prior organ allograft or allogenic bone marrow transplantation Have had organ allograft or hematopoietic transplantation. Subject has a history of organ allograft including liver transplant. Subjects with organ allograft requiring immunosuppression Organ transplant (allograft) recipient Organ transplant (allograft) recipient Prior organ allograft or allogeneic transplantation, if the transplanted tissue is still in place Patients must not have an organ allograft or other history of immune compromise Subjects with organ allograft requiring immunosuppression Presence of any tissue or organ allograft, regardless of need for immunosuppression, including corneal allograft; patients with a history of allogeneic hematopoeitic stem cell transplant will be excluded Subjects with organ allograft requiring immunosuppression History of organ allograft History of organ allograft Prior splenectomy or organ allograft History of organ allograft Patient with an organ allograft History of organ allograft History of organ allograft. Organ transplant (allograft) recipient reconstruction to include structural allograft; Corneal defects, corneal ulcerations, keratitis, keratoconus, history of corneal transplant, or other known abnormalities of the cornea that may pose an increased risk of developing a corneal ulcer History of organ allograft (including corneal transplant) History of organ allograft including corneal transplant History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) History of confirmed, corneal ulceration History of corneal transplantation Corneal disease, such as bullous or band keratopathy, corneal desquamation, keratitis, corneal ulcer, or keratoconjunctivitis History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) except stem cell transplants History of corneal transplantation History of organ allograft (including corneal transplant) REGORAFENIB EXCLUSION CRITERIA: History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant). History of organ allograft including corneal transplant History of solid organ transplantation (with the exception of a corneal transplant > 3 months prior to screening) History of organ allograft (including corneal transplant) History of organ allograft including corneal transplant History of organ allograft (including corneal transplant) Corneal defects, corneal ulcerations, keratitis, keratoconus, history of corneal transplant, or other known abnormalities of the cornea History of corneal transplantation History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant). History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) History of organ allograft. (Including corneal transplant) History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) Corneal pathology that would limit evaluation of loss in visual acuity associated with corneal deposits. History of organ allograft (including corneal transplant). History of organ allograft (including corneal transplant) History of corneal disease Active corneal erosions or history of abnormal corneal sensitivity test History of solid organ transplantation (with the exception of a corneal transplant > 3 months prior to screening). History of organ allograft (including corneal transplant) History of organ allograft (including corneal transplant) Current corneal disease or a history of corneal disease. Have prior allogeneic bone marrow transplantation or solid organ transplant Prior treatment with any anti-PD-1 or anti-PD-L1 antibody, prior allogeneic bone marrow transplantation, or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients must not have undergone prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients must not have undergone prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation, double umbilical cord blood transplantation (dUCBT) or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation 7. Prior allogeneic bone marrow transplantation or prior solid organ transplantation; Prior organ allograft or allogeneic bone marrow transplantation Has had any prior allogeneic solid organ or bone marrow transplantation. Prior allogeneic SCT or solid organ transplantation Prior organ transplantation or allogeneic bone marrow transplantation. Prior allogeneic or autologous bone marrow or organ transplantation Prior allogeneic or autologous bone marrow transplantation or other solid organ transplantation. Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow transplantation or solid organ transplant for another malignancy in the past Participants must not have had prior organ allograft or allogeneic bone marrow transplantation MEDICATION-RELATED: Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation. Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior organ allograft or allogeneic bone marrow transplantation Prior allogeneic bone marrow transplantation or prior solid organ transplantation. (Atezolizumab-related exclusion) Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with a prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation No prior allogeneic bone marrow or organ transplantation Patients with a prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation Patient has history of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic or autologous bone marrow transplantation or other solid organ transplantation. Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation. Prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Subjects who have had solid organ or bone marrow transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Participants with prior bone marrow or solid organ transplantation. Prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or solid organ transplantation Participants with prior allogeneic bone marrow transplantation or prior solid organ transplantation Participants with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patient has a history of hematological malignancy within the last 5 years prior to study entry; prior allogeneic bone marrow transplantation or prior solid organ transplantation Had solid organ or bone marrow transplantation Prior allogeneic bone marrow or organ transplantation Prior allogeneic bone marrow transplantation or solid organ transplant Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow transplantation or prior solid organ transplantation Subjects who have had solid organ or bone marrow Transplantation. Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation. Prior allogeneic bone marrow or organ transplantation Prior allogeneic bone marrow or organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation Prior allogeneic bone marrow or organ transplantation Patient must not have had a prior allogeneic bone marrow or organ transplantation Prior allogeneic bone marrow or organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Patients with prior allogeneic solid organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation Prior allogeneic bone marrow or organ Prior bone marrow or organ transplantation. Prior allogeneic bone marrow or organ transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or solid organ transplantation Prior allogeneic bone marrow or organ transplantation Subjects may not have had prior organ allograft or allogeneic bone marrow transplantation Prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior organ allograft or allogeneic bone marrow transplantation Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogeneic bone marrow or organ transplantation Prior allogeneic or autologous bone marrow transplantation or other solid organ transplantation. Participants with prior allogeneic bone marrow transplantation or prior solid organ transplantation Prior allogenic bone marrow transplantation or prior solid organ transplantation Patients may not have had a prior allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) Patients must not have had prior allogeneic bone marrow transplant or double umbilical cord blood transplantation Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) 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-1 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 4 of 6 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: Each UCB unit MUST contain at least 1.5 x 10^7 TNC per kilogram recipient weight DONOR: Unrelated Umbilical Cord Blood: The total cell dose of the combined units must be at least 3.0 x 10^7 TNC per kilogram recipient weight DONOR: Unrelated Umbilical Cord Blood: Any cord blood units with < 1.5 x 10^7 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-1 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 Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tositumomab (Bexxar), as part of their salvage therapy are not eligible for myeloablative umbilical cord blood transplant Previous allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUCBT). Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) 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 5/8 HLA or better high resolution matched umbilical cord blood (UCB) grafts with a cell dose of 2.0 x 10^7 total nucleated cell (TNC)/kg each, or\r\n* A related haplo-identical donor Previous allogenic bone marrow transplant or double umbilical cord blood transplantation Patients must have two cord blood (CB) units available which are matched with the patient at 4, 5, or 6/6 HLA\r\nclass I (serological) and II (molecular) antigens. Each cord must contain at least 1.5 x 10^7 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. 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 Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) Previous allogenic bone marrow transplant or double umbilical cord blood transplantation UCB graft selected according to current University of Minnesota umbilical cord blood graft selection algorithm No prior allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUBCT) 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 2 cord blood (CB) units may be used to achieve the required cell dose PART 2: Patients without an HLA-identical or 1-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 DONOR (FHCRC only): Up to 5% 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 < 1.5 x 10^7 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)-1 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 Patients who have received < 2 cycles of multiagent chemotherapy and patients who have received no multiagent chemotherapy within the 3 months previous to umbilical cord blood transplant (UCBT) as well as patients experiencing graft failure following previous allogeneic transplant DONOR: Any cord blood units with < 1.5 x 10^7 total nucleated cells per kilogram recipient weight 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 Children’s 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 > 2.0 x 10^7 cells/kg and will be matched at 4 - 6 of 6 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) (> 7/8 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., > 1.5 x 10^7/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 Patients who have received Y-90 ibritumomab (zevalin) or I-131 tostumomab (bexxar), as part of their salvage therapy are not eligible for myeloablative umbilical cord blood transplant UCB units will be selected according to current University of Minnesota umbilical cord blood graft selection algorithm; one or 2 UCB units may be used to achieve the required cell dose Previous allogenic bone marrow transplant or cord blood transplantation. Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) 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 Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT). Previous allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUCBT) Evidence suggestive of myelodysplastic syndrome (MDS) or acute myelogenous leukemia (AML) on peripheral blood smear or bone marrow biopsy, if clinically indicated\r\n* No prior allogeneic bone marrow transplant or double umbilical cord blood transplantation (dUBCT) Patients who have a 5/6 or better matched related donor or a 4/6 or better umbilical cord blood donor and who are medically eligible for conventional myeloablative or nonmyeloablative transplant will be excluded 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. No prior allogenic bone marrow transplant or double umbilical cord blood transplantation Patients receiving bone marrow or umbilical cord blood as a stem cell source may also be considered for enrollment with acknowledgement that if there is insufficient product available for DLI, the patient will receive azacitidine without DLI per standard-risk treatment 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 Previous allogenic bone marrow transplant or double umbilical cord blood transplantation (dUCBT). 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 Umbilical cord HCT Eligible patients will have one or more of the following donor stem cell sources:\r\n* Bone marrow\r\n* Placental blood (umbilical cord blood)\r\n* Cytokine mobilized peripheral blood 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 4-6/6 to the patient, considering HLA-A, B at the antigen level and DRB1 at the allele level Prior allogeneic bone marrow transplant or double umbilical cord blood transplantation Receiving first allogeneic HSCT (bone marrow, peripheral blood stem cell, or cord blood transplant) Unrelated donor, umbilical cord blood, mismatched, or haploidentical transplants