Inclusion Criteria:

          1. Presence of high-risk acute leukemia/myelodysplasia defined as one of the following:

             I. Acute Myeloid Leukemia:

               1. Primary induction failure (no CR or CRi after ≥ 2 courses of induction therapy or
                  after ≥ 1 induction containing high dose Ara-C)

               2. Chemorefractory relapse (no CR or CRi after 1 chemointensive treatment)

               3. Relapse after allogeneic or autologous transplant

               4. High risk AML in CR1: i) any adverse genetic abnormality as defined by European
                  Leukemia Net excluding FLT3 mutation; ii) secondary or therapy related AML
                  excluding good risk genetic abnormalities (as defined by ELN); or iii) any other
                  poor risk feature known to be associated with a PFS or DFS ≤40% at 2 years after
                  conventional transplantation.

               5. CR2 excluding good risk genetic abnormalities defined by ELN

               6. ≥CR3

             II. Acute Lymphoid Leukemia

               1. Primary induction failure (≥ 2 inductions)

               2. Chemorefractory relapse (at least 1 intensive induction chemotherapy;
                  blinatumomab, inotuzumab or CAR-T cells may be considered as an equivalent)

               3. Relapse after allogeneic or autologous transplant

               4. High risk ALL in CR1: Ph like ALL or any other poor risk feature known to be
                  associated with an PFS or DFS ≤40% at 2 years after conventional transplantation.

               5. ≥CR2

               6. MRD+ within 1 month of start of conditioning regimen.

             III. Myelodysplastic syndrome

               1. Relapse after allogeneic or autologous transplant

               2. ≥10 % blasts within 1 month of start of conditioning regimen

               3. Very poor cytogenetics (>3 abnormalities)

               4. Any poor risk feature known to be associated with a PFS or DFS ≤40% at 2 years
                  after conventional transplantation

               5. TP53 mutation

               6. ≥40 years old and RAS or JAK2 mutation

               7. CMML with HCT-specific CPSS score high or intermediate-2

               8. Stable disease (absence of CR/PR/HI) after 6 cycles of azacitidine (or another
                  demethylating agent)

               9. Progressive disease while on azacitidine (or another demethylating agent)

          2. 18-70 years old

          3. Availability of 2 CBs ≥ 4/8 HLA match when A, B, C and DRB1 are performed at the
             allele level.

             I. Cord to be expanded:

               1. CD34+ cell count >0.5 x 105/kg and TNC>1.5 x 107/kg (these numbers are all
                  pre-freeze)

               2. Needs to be erythrodepleted by bank prior to cryopreservation

               3. Must come from a cord bank that is FACT (Foundation for the Accreditation of
                  Cellular Therapy) accredited, FDA approved or eligible for NMDP IND.

             II. Non-expanded CB/back-up cord:

               1. Pre-freeze TNC count ≥ 2.0 x 107/kg with CD34+ cells ≥1.5 x 105/kg or TNC count ≥
                  1.5 x 107 TNC/kg with CD34+ cells ≥1.7 x 105/kg. If a single cord does not meet
                  these criteria, 2 back up cords will be an acceptable alternative with a minimum
                  for each of 1.5 x 107/kg TNC and 1 x 105/kg CD34+ cells; another acceptable HSC
                  back up source could be a haploidentical donor with medical clearance prior to
                  starting conditioning regimen.

               2. Must come from a cord bank that is FACT accredited, FDA approved or eligible for
                  NMDP IND

          4. Karnofsky score ≥ 70%

          5. Bilirubin < 2 x upper limit of normal (ULN) unless felt to be related to Gilbert's
             disease or hemolysis; AST and ALT ≤ 2.5 x ULN; alkaline phosphatase ≤ 5 x ULN.

          6. Estimated or measured creatinine clearance ≥ 60 ml/min/1.73m2.

          7. Hematopoietic cell transplantation specific comorbidity index (HCT-CI) ≤5 for patients
             < 60 years old; HCT-CI ≤3 for patients < 60 years old and acute leukemia not in
             CR/CRi; HCT-CI ≤3 for patients 60-65 years old; HCT-CI ≤1 if 66-70 years old.

          8. Left ventricular ejection fraction ≥ 40%

          9. Forced vital capacity (FVC), forced expiratory volume in 1 second (FEV1) and diffusing
             capacity corrected for hemoglobin (DLCOc) ≥ 50% of predicted

         10. Signed written informed consent

         11. Female patients of childbearing potential must have a negative serum pregnancy test
             within 7 days of enrolment and must be willing to use an effective contraceptive
             method while enrolled in the study.

        Exclusion Criteria:

          1. Patient never treated with cytotoxic chemotherapy and planned conditioning regimen
             does not include 12 Gy TBI (exceptions allowed if approved by PI).

          2. Allogeneic myeloablative transplant within 6 months.

          3. Autologous hematopoietic stem cell transplant within 6 months.

          4. Planned use of ATG in conditioning regimen (exceptions allowed if approved by PI in
             which case ATG must be adjusted for weight/lymphocyte count and given more than 1 week
             prior to transplant; any patient who receives ATG will have immune recovery studies
             but will not be counted with rest of patients and will be analyzed separately).

          5. Planned use of an HLA matched CB (8/8 allele matched)

          6. Uncontrolled infection.

          7. Presence of a malignancy other than the one for which the CB transplant is being
             performed, with an expected survival estimated to be less than 75% at 5 years.

          8. Seropositivity for HIV.

          9. Hepatitis B or C infection with measurable viral load. Patients with chronic hepatitis
             B or C infection regardless of viral load require clear documentation of absence of
             cirrhosis by either fibroscan or biopsy. If fibroscan is the method used, the test
             must be unequivocally negative.

         10. Liver cirrhosis.

         11. Active central nervous system involvement

         12. Chloroma > 2 cm

         13. ≥50% blasts in marrow in an evaluable marrow sample (>25% of normal cellularity for
             age) collected less than one month prior to start of conditioning regimen.

         14. Peripheral blasts >1000/mm3

         15. Pregnancy, breastfeeding or unwillingness to use appropriate contraception.

         16. Participation in a trial with an investigational agent within 30 days prior to entry
             in the study.

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

         18. Any abnormal condition or laboratory result that is considered by the PI capable of
             altering patient's condition or study outcome.