Intent to proceed with RIC alloHCT if a matched sibling or matched unrelated donor is identified. There is no requirement as to the timing of the transplantation. A suitable donor must be identified; there are no restrictions on donor type and can include a matched sibling, a matched or mismatched unrelated donor, a family haplotype matched donor or a cord blood donor (single or double) The HLA-matched donor must be medically fit to donate and willing to donate bone marrow. DONOR: Matched unrelated donors\r\n* Unrelated volunteer donor matched for HLA-A, -B, -C and -DRB1 defined by high resolution molecular typing\r\n** Mismatched unrelated volunteer donors may be considered if no other suitable donor is available Any identified and available 10/10 HLA-matched related donor or 10/10 HLA-matched unrelated donor. Available mismatched related (mMRD) or mismatched unrelated (mMUD) donor, HLA matched 8/10 or 9/10 HLA-matched related or unrelated donor available Patients with suitably matched related or unrelated donor, as defined per institutional practice. Patients who have a matched donor and are candidates for allogeneic bone marrow transplantation Patient with a HLA-matched (HLA-A, B, C, and DR beta 1 [DRB1] molecularly matched) unrelated donor or related donor capable of donating PBSC Patients with Hodgkin Lymphoma with either of the following: • Primary induction failure (failure to achieve initial CR) and/or primary refractory disease OR First, Second or Third relapse AND History of prior ablative auto HSCT or ineligible for an ablative auto HSCT or ?25% residual disease after at least two reinduction chemotherapy cycles AND HLA matched family donor (6/6 or 5/6) or matched unrelated adult donor (MUD) (8/8) or matched umbilical cord blood unit (?5/6) with prethaw cell dose of at least 3 x 107/kg TNC. No identified 8/8 (based upon A, B, C, DRB1 loci) allele matched unrelated donor, or unable to wait sufficient time to procure an 8/8 allele matched unrelated donor DONOR: Arm B: The recipient must have a related donor genotypically HLA-A, B, C and DRB1 loci haploidentical to the recipient; no HLA matched sibling or matched unrelated donor is available; DSA is allowed with desensitization done if recommended by donor selection committee (DSC) per City of Hope (COH) standard operating procedures (SOP) Patients with no available and suitably matched related or unrelated donor in the required time period. Patients without a matched related or unrelated donor Patient must have planned to receive either a myeloablative or reduced-intensity conditioning regimen and have an unrelated donor who is HLA matched or single-allele mismatched Patient with hematologic malignancy not in CR: Does NOT have a 10/12 (or better) HLA matched related or unrelated donor available OR treating clinician considers haploidentical transplant preferable (despite sibling donor availability or 10/12 or better HLA matched donor availability) due to patient’s clinical status HLA-MATCHED UNRELATED DONOR: Patient and donor pairs homozygous at a mismatched allele in the graft rejection vector are considered a two-allele mismatch, i.e., the patient is A*0101 and the donor is A*0102, and this type of mismatch is not allowed Meets institutional criteria for a RIC allogeneic (allo) BMT to treat a B?cell derived hematologic malignancy of a fully matched or partially mismatched related or unrelated donor source Lack of an HLA matched donor or need to proceed fast to transplantation when a patient does not have an immediately available matched unrelated donor (typed by high-resolution in the registry) Patient must not have a 10/10 HLA matched family member or unrelated donor. Subjects have undergone alloSCT > 90 days prior to enrollment from a matched-related donor (MRD), matched-unrelated donor (MUD), cord blood donor, or haplo-identical and cord blood donor DONOR: HLA 6/6 matched related or unrelated donor Patients must have a haploidentical related donor or a fully matched related or unrelated donor Subjects must have a suitable stem cell donor available who may donate cells if the subject needs to undergo allogeneic hematopoietic cell transplant (HCT); donor may be matched or mismatched and must be found to be suitable according to the institution’s standard criteria HLA-matched stem cell donor, either related (6/6 or 5/6 loci matched) or unrelated (8/8 or 7/8 loci matched) \r\n* Note: unrelated donors should be matched at HLA-A, -B, -C, and -DRB1 loci; however, a single locus mismatch will be acceptable in the event a more closely matched donor is not available Patients with hematologic diseases who have undergone T-cell depleted reduced intensity/non-ablative allogeneic transplantation, using a 7-8/8 HLA-matched related or unrelated donor or 4-6/8 HLA-matched related donor; this may include patients with a mixed chimeric state or disease persistence or at high risk of relapse HLA 4-8/8 MATCHED RELATED DONOR Will be matched at least as HLA -A, -B, C and -DRB1; criterion for donation will be those allowing donation following the National Marrow Donor Program (NMDP) accepted donor criterion and program standard operating procedures (SOPs) for the typical matched unrelated donors No HLA matched sibling or matched unrelated donor is available Donor criteria: availability of a donor either an human leukocyte antigen (HLA) matched sibling donor (MSD) or a haploidentical (5-9/10 HLA matched); alternatively a 8/8 HLA matched unrelated donor (MUD) by high resolution typing is immediately available Donor source is matched related, unrelated, haploidentical donor or cord blood. DONOR: 10 of 10 HLA-matched or 1 allele mismatched (9 of 10) unrelated donor Available familial haploidentical (4 to 6 out of 8 HLA loci-matched) donor Donor is either matched related, matched unrelated, mismatched unrelated, or haploidentical; cord blood recipients are also eligible HLA-matched donor able to donate Patients must have an allogeneic hematopoietic progenitor cell donor (HPCT), either a matched sibling, mismatched (1 allele) sibling, or a matched unrelated donor (MUD) or a mismatched (1 allele) unrelated donor\r\n* Previous autologous hematopoietic progenitor cell transplantation is allowed; a minimum of 6 months should have elapsed from prior autologous hematopoietic progenitor cell transplantation; prior transplantation with conditioning regimens using total body irradiation is not allowed Haploidentical Related Donor Selection Criteria: the recipient must have a related donor haploidentical for HLA A, B, C, and DR (i.e. matched at one locus); they may be partially matched on the other haplotype; if a matched sibling donor is available but disease status (poor disease control, recurrence after allogeneic HSCT) precludes treatment on other protocols, a haplo-identical relative is preferred as the HSC (cluster of differentiation [CD]34+ cell) source RECIPIENT: No available 10/10 or 9/10 HLA-matched related or unrelated donor, 4/6 (or greater) matched UCB unit(s) with a total dose of greater than or equal to 3.5 x 10^7 TNC/kg, or haploidentical related donor MATCHED RELATED DONOR: Ability to give informed consent MATCHED RELATED DONOR: History of other medical conditions that in the opinion of PI constitute a contraindication to donation MATCHED UNRELATED DONOR: Failure to qualify as an NMDP donor Patients must have a fully-matched sibling donor or a matched unrelated donor identified Patients must have an HLA-matched donor of hematopoietic stem cells (related or unrelated) DONOR: Patient and donor (related or unrelated) must be HLA-matched and express HLA-A*0201 HLA-matched related donor able to donate Absence of a suitable related or unrelated bone marrow donor who is molecularly matched at HLA-A, B, Cw, DRB1, and DQB1 No available 10/10 or 9/10 HLA-matched related or unrelated donor or haploidentical related donor MATCHED RELATED DONOR: Age 2-60 years, and weight of >= 10 kilograms MATCHED RELATED DONOR: At least one normal DOCK8 allele demonstrated by a CLIA-certified lab MATCHED RELATED DONOR: History of severe cutaneous viral infections with herpes simplex, herpes zoster, or molluscum contagiosum MATCHED RELATED DONOR: HIV infection MATCHED RELATED DONOR: Chronic active hepatitis B; donor may be hepatitis core antibody positive MATCHED RELATED DONOR: Other medical conditions that in the opinion of the PI constitute exclusion as a donor MATCHED RELATED DONOR: Mutation of DOCK8 on both alleles MATCHED UNRELATED DONOR: Failure to qualify as an NMDP donor HLA-matched or single allele-mismatched donor able to donate Cross-over to other tandem autologous-allogeneic research protocol (#1409 or other appropriate protocol) will be allowed if a suitable HLA-matched related or unrelated donor is identified before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study Cross-over from other tandem autologous-allogeneic research protocol (#1409 or other appropriate protocol) will be allowed if the patient loses the suitable HLA-matched related or unrelated donor but has an available HLA-haploidentical donor before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study Available matched related or unrelated donor; selected donor must be a complete match or have only a single antigen mismatch No available and suitably matched related or unrelated donor Very well-matched related or unrelated donor DONOR: An unrelated donor–fully matched DONOR: An unrelated donor–9/10 matched Patients who have a matched related donor who is eligible and willing to donate stem cells All patients, irrespective of participation in a clinical trial, must have a signed informed consent for the performance of matched related donor transplantation Absence of timely and suitable fully HLA matched or one HLA locus mismatched family or unrelated donor and, at Investigator's discretion, absence of other possible therapeutic alternatives Patients must have resolved any serious infectious complications related to induction\r\n* NOTE: Patients with an HLA-matched donor and proceeding to transplant will be allowed up to one cycle of consolidation treatment No availability of a readily available HLA-matched volunteer unrelated donor (8 of 8 allele match at HLA-A, -B, -C and -DRB1); patients with unstable disease who are in danger of significant disease progression while waiting to procure volunteer donor cells will be eligible to be treated on this protocol, even if a matched donor is available Availability of a suitable matched related (6/6 or 5/6) or unrelated donor (10/10 or 9/10 antigen or allele matched) Is undergoing matched or single-antigen mismatched unrelated-donor myeloablative transplant for the treatment of ALL or AML; Is less than or equal to (<=) 60 years of age For the cohort after Recommended phase 2 dose (RP2D) Is undergoing matched or single-antigen mismatched related or unrelated-donor transplant and receiving myeloablative conditioning or RIC for the treatment of hematologic malignancies or myeloproliferative neoplasms; Is less than or equal to (<=) 75 years of age Matched Related Donor: Related to the patient and is genotypically or phenotypically HLA-identical Matched Related Donor: Donor age < 75 yrs unless cleared by institutional PI Matched Related Donor: Donor must consent to PBSC mobilization with G-CSF and apheresis HLA Matched Unrelated Donor: Donor must consent to PBSC mobilization with G-CSF and apheresis; bone marrow unrelated donors are not eligible for this protocol Matched Related Donor: Identical twin Matched Related Donor: Any contra-indication to the administration of subcutaneous G-CSF at a dose of 16mg/kg/d for five consecutive days Matched Related Donor: Serious medical or psychological illness Matched Related Donor: Pregnant or lactating females Matched Related Donor: HIV seropositivity Known HLA matched related donor without contraindications to donate HLA-matched donor able to donate HLA-matched related or 8/8 allele matched (HLA-A, -B, -C, -DRB1) unrelated donor available. This exclusion does not apply to HIV-positive subjects who have a CCR5delta32 homozygous donor. Subjects should have a potential 3-5/6 HLA-matched related haploidentical donor that will be evaluated for eligibility to provide DLI DONOR: Patients must have a healthy HLA matched or mismatched related or unrelated donor who is willing to receive filgrastim (G-CSF) injections and undergo apheresis for peripheral blood stem cell (PBSC) collection, or undergo a marrow harvesting procedure HLA-Matched Related and Unrelated Donors: Patients who have an HLA-matched related or unrelated donor are eligible for entry on this protocol; this will include a healthy donor who is genotypically matched at all A, B, C, DRB1 and DQB1 loci, as tested by deoxyribonucleic acid (DNA) analysis Patients must have a fully-matched related donor or a matched unrelated donor identified; double cord (at least 4/6 matched) can be used if no adult matched donor is available MATCHED RELATED DONOR: History of medical illness that in the estimation of the PI or Department of Transfusion Medicine (DTM) physician precludes donation of marrow MATCHED RELATED DONOR: Pregnant females MATCHED RELATED DONOR: Any medical contraindication to anesthesia or marrow donation will exclude the donor MATCHED RELATED DONOR: Active autoimmune disease that in the opinion of the PI or AI would compromise the success of the transplant HLA-matched related and unrelated donors: \r\n* Patients who have an HLA-matched related or unrelated donor are eligible for entry on this protocol; this will include a healthy donor who is genotypically matched at all A, B, C, DRB1 and DQB1 loci, as tested by deoxyribonucleic acid (DNA) analysis Patients must have a 6/6 HLA-matched related donor who is evaluated and deemed able to provide peripheral blood stem cells (PBSCs) and/or marrow by the transplant team DONOR: Donor must be 6/6 HLA matched, and related to the patient Patients for whom HLA-matched unrelated donor search could not be initiated or completed due to insurance reasons, concerns of rapidly progressive disease, and/or discretion of attending physician are eligible for this protocol No available suitably HLA- matched unrelated donor Available suitably HLA- matched unrelated donor is unwilling to donate peripheral blood stem cells (PBSC), and no alternate donor is found Availability of appropriate, willing, HLA-matched related marrow donor Matched related or unrelated donor identified and available; donor must be a complete match or have only a single allele or antigen mismatch Must have an HLA genotypically or phenotypically identical related donor or, at a minimum, a high likelihood of identifying an HLA-matched unrelated donor; the determination of availability of a suitable unrelated donor may be based on a World-Book search Cross-over to other tandem autologous-allogeneic research protocol (#2241) will be allowed if the patient loses the suitable HLA-matched related or unrelated donor but has an available HLA-haploidentical donor before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study Cross-over from other tandem autologous-allogeneic research protocol (#2241) will be allowed if a suitable HLA-matched related or unrelated donor is identified before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study Patient must have a fully matched related or unrelated donor willing to donate stem cells Must have suitable matched sibling or matched unrelated donor for stem cell source For Pre-allo Part A (before stem cell transplant): Availability of an HLA matched related or unrelated donor Lack of 5 6/6 HLA matched related or 8/8 HLA A, B, C, DRß1 matched unrelated donor; or unrelated donor not available within appropriate timeframe, as determined by the transplant physician. HLA-matched related donor able to donate. HLA-matched donor able to donate Patient should have an already identified sibling, matched unrelated donor or cord blood donor at the time of enrollment to this clinical trial No available suitable matched related (6/6 or 5/6) or unrelated donor (8/8 or 7/8 allele matched) or unwilling or unable to pursue allogeneic stem cell transplant Does not have a suitable matched related/sibling donor (MSD) or volunteer matched unrelated donor (MUD) available in the necessary time for stem cell donation Subjects with who have undergone a non-myeloablative allogeneic transplant, using a 3-6/6 HLA matched related donor Patients < 55 years (myeloablative regimen #1) or > 55 and =< 75 years or significant comorbidities (reduced intensity regimen #2) old lacking a matched related volunteer donor identified in time for transplant for which a related haploidentical donor (=< 7/8 allele match at the A, B, C, DR loci), a 7/8 allele matched related or unrelated donor is identified, or a matched unrelated donor (MUD); the patients must be diagnosed with high-risk disease as following: Matched unrelated donor must consent to provide a marrow allograft Matched (8/8) or mismatched (7/8) related, unrelated HCT Patient has a suitable and willing HLA-8/8 matched or 7/8 mismatched (at one allele) unrelated donor identified Patients must have a related or unrelated donor as follows: Availability of a suitable 8/8 HLA-A, -B, -C, and -DRB1-matched unrelated mPB donor; Patients who have undergone allogeneic HSCT in CR1 from a matched related or matched unrelated donor. All of the following criteria must also be met: Histocompatible donor identified:\r\n* Related donor or unrelated donor matched 5/6 or better (A, B, DRB1) Available matched unrelated donor Donor choices per matched unrelated donor (MUD) committee according to center standard operating procedure (SOP) Recipients of 8-7/8 HLA-matched donor; post-HSCT period within day +100 to day +150 Patients must have an available 8/8 HLA-A, -B, -C, and -DRB1 matched-related or unrelated donor allogeneic hematopoietic peripheral blood stem cell graft Planned recipient of a first kidney allograft from an HLA-matched, living related donor Patient plans on receiving stem cells from a matched (8/8) related donor Clinical justification of allogeneic stem cell transplantation where a suitable HLA matched sibling or unrelated donor is unavailable in a timely manner. An unrelated donor search is not required for a patient to be eligible if the clinical situation dictates an urgent transplantation. Clinical urgency is defined as 6-8 weeks from referral to transplant center or low likelihood of finding a matched unrelated donor Availability of a fully matched related or unrelated donor following a donor search HLA-matched, related or 7-or 8/8 allele matched (HLA-A,-B,-Cw,-DRBl) unrelated donor able to donate.