Participant Inclusion Criteria:

        Age and Donor Status:

        Patients with age ≤ 21 years at time of consent with no available and suitably matched
        related or unrelated donor in the required time period.

        Diagnoses :

        I. Acute myelogenous leukemia (AML) :

          -  Complete first remission (CR1) at high risk for relapse such as any of the following:

               -  Known prior diagnosis of myelodysplasia (MDS) or myeloproliferative disorder
                  (MPS).

               -  Therapy-related AML (t-AML).

               -  White cell count at presentation > 100,000.

               -  Presence of extramedullary leukemia at diagnosis.

               -  Any unfavorable subtype by FAB or WHO classification.

               -  High-risk cytogenetics (e.g. those associated with MDS, abnormalities of 5, 7, 8,
                  complex karyotype) or high-risk molecular abnormalities.

               -  Requirement for 2 or more inductions to achieve CR1.

               -  Presence of Minimal Residual Disease (MRD+) by cytogenetics, flow cytometry or
                  molecular methods after induction.

               -  Any patient with newly diagnosed AML with intermediate risk cytogenetics who
                  elects allograft with curative intent over consolidation chemotherapy.

               -  Any patient unable to tolerate consolidation chemotherapy as would have been
                  deemed appropriate by the treating physician.

               -  Other high-risk features not defined above.

          -  Complete second remission (CR2).

          -  Primary refractory or relapsed AML with less than 10% blasts by bone marrow
             morphology. Patients with cytogenetic, flow cytometric, or molecular abnormalities in
             ≤ 10% of cells are eligible.

        II. Acute lymphoblastic leukemia (ALL):

          -  Complete first remission (CR1) at high risk for relapse such as any of the following:

               -  White cell count at presentation > 30,000 for B-cell lineage and > 100,000 for
                  T-cell lineage.

               -  Presence of any high-risk cytogenetic abnormalities such as t (9;22), t (1;19), t
                  (4;11) or other MLL rearrangements (11q23) or other high-risk molecular
                  abnormality.

               -  Failure to achieve complete remission (CR) after four weeks of induction therapy.

               -  Persistence or recurrence of MRD on therapy.

               -  Any patient unable to tolerate consolidation and/or maintenance chemotherapy as
                  would have been deemed appropriate by the treating physician.

               -  Other high-risk features not defined above.

          -  Complete second remission (CR2).

          -  Primary refractory or relapsed ALL with MRD disease after antibody therapy (e.g.,
             blinatumomab, inotuzumab, other) and/or CAR-T cell therapy.

        III. Other acute leukemias:

        Leukemias of ambiguous lineage or of other types (e.g. blastic plasmacytoid dendritic cell
        neoplasm) with less than 5% blasts by BM morphology. Patients with persistent/relapsed
        disease with cytogenetic, flow cytometric or molecular aberrations in ≤ 5% of cells are
        eligible.

        IV. Myelodysplastic Syndrome (MDS) / Myeloproliferative Disorders (MPD) other than
        myelofibrosis:

          -  International prognostic scoring system (IPSS) risk score of INT-2 or high risk at the
             time of diagnosis.

          -  Any IPSS risk category if life-threatening cytopenia(s) exists.

          -  Any IPSS risk category with karyotype or genomic changes that indicate high risk for
             progression to acute myelogenous leukemia.

          -  MDS/ myeloproliferative disorder overlap syndromes without myelofibrosis.

          -  MDS/ MPD patients must have less than 10% bone marrow myeloblasts and ANC ≥ 0.2
             (growth factor supported if necessary) at transplant work-up.

        V. Non-Hodgkin lymphoma (NHL) or Hodgkin lymphoma (HL) at high-risk of relapse or
        progression if not in remission:

          -  Eligible patients with aggressive histology (such as, but not limited to, diffuse
             large B-cell NHL, mantle cell NHL, and T-cell histology) in CR.

          -  Eligible patients with indolent B cell NHL (such as, but not limited to, follicular,
             small cell or marginal zone NHL) will have 2nd or subsequent progression with stable
             disease/ CR/ PR with no single lesion equal to or more than 5 cm.

          -  Eligible patients with HL will be those without progression of disease (POD) after
             salvage chemotherapy with no single lesion ≥ 5 cm.

        VI. Inherited Metabolic Disorders [also see EBMT Handbook for discussion on patient
        eligibility for allogeneic transplant; in general, patients are considered early in the
        disease course, before they develop neurologic symptoms (46)]:

          -  Hurler Syndrome

          -  Hunter (MPS 2 - early disease)

          -  Sly syndrome (MPSVIII)

          -  α-Mannosidosis

          -  X- ALD

          -  Osteopetrosis

          -  Metachromatic Leukodystrophy

          -  Globoid (GLD)

        VII. Non-Malignant disorders (other) [also see EBMT Handbook for criteria for transplant
        (46)]

          -  Hemoglobinopathies

          -  Bone Marrow Failure syndromes

          -  Immunodeficiencies, including HLH

        Organ Function and Performance Status Criteria:

          -  Karnofsky or Lansky score ≥ 70% (see Appendix)

          -  Bilirubin ≤ 1.5 mg/dL (unless benign congenital hyperbilirubinemia).

          -  ALT ≤ 3 x upper limit of normal.

          -  Pulmonary function (spirometry and corrected DLCO) ≥ 50% predicted (corrected for
             hemoglobin) .

          -  Left ventricular ejection fraction ≥ 50%.

          -  Age-adjusted Hematopoietic Cell Transplantation-Comorbidity Index (aaHCT-CI) less than
             or equal to 7.

          -  Renal: serum creatinine ≤ 1.5x normal for age. If serum creatinine is outside the
             normal range, then CrCl > 50 mL/min/1.73m2 (calculated or estimated) or GFR
             (mL/min/1.72m2) >30% of predicted normal for age.

        Normal GFR in Children and Young Adults (Age) : Mean GFR +- SD (mL/min/1.73 m2)

        1 week: 40.6 + / - 14.8 2-8 weeks: 65.8 + / - 24.8 >8 weeks: 95.7 + / - 21.7 2-12 years:
        133.0 + / - 27.0 13-21 years (males): 140.0 + / - 30.0 13-21 years (females: 126.0 + / -
        22.0

        GFR, glomerular filtration rate; SD, standard deviation greater than 2 years old: Normal
        GFR is 100 mL/ min. Infants: GFR must be corrected for body surf ace area.

        For metabolic diseases: disease status to be evaluated according to EBMT Handbook [45].

        Graft Criteria

        CB units will be selected according to the current MSKCC unit selection algorithm. High
        resolution 8 allele HLA typing and recipient HLA antibody profile will be performed. Cord
        unit selection will occur based on HLA-match, total nucleated cell (TNC) and CD34+ cell
        dose adjusted per patient body weight. The cord bank of origin will also be considered.
        Donor specific HLA antibodies, if present, will also be taken into consideration and may
        influence the selection of the graft. CB graft will consist of one or two CB units (CBU)
        based on MSKCC selection algorithm.

          -  Each CB unit must be at least 3/8 HLA-matched to the patient considering
             high-resolution 8-allele HLA typing.

          -  For malignant diseases follow MSKCC CBU selection algorithm

          -  For non-malignant diseases, CBU will be required to have > 5 x 107 TNC/kg; high HLA
             allele level match is preferable

        Participant Exclusion Criteria:

          -  Inadequate performance status/ organ function.

          -  Advanced metabolic disease (EBMT handbook).

          -  Active CNS leukemic involvement.

          -  Indolent NHL or Hodgkin lymphoma with progression of disease after most recent salvage
             chemotherapy.

          -  Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow
             fibrosis.

          -  Autologous stem cell transplant within the preceding 6 months.

          -  Any prior allogeneic stem cell transplant.

          -  Active and uncontrolled infection (bacterial/fungal/viral) at time of transplantation.

          -  HIV infection.

          -  Seropositivity for HTLV-1.

          -  Pregnancy or breast feeding.

          -  Patient or guardian unable to give informed consent or unable to comply with the
             treatment protocol including appropriate supportive care, long-term follow-up, and
             research tests.