Inclusion Criteria:

          -  Patients with the following hematologic malignancies:

               -  Acute myelogenous leukemia (AML): High-risk AML including:

                    -  Antecedent hematological disease (e.g., myelodysplasia (MDS))

                    -  Treatment-related leukemia

                    -  Complete Remission (CR1) with poor-risk cytogenetics or molecular markers
                       (e.g. Flt 3 mutation, 11q23, del 5, del 7, complex cytogenetics)

                    -  Second complete remission (CR2) or third complete remission (CR3)

                    -  Induction failure or 1st relapse with ≤ 10% blasts in the marrow

               -  Acute lymphoblastic leukemia (ALL)

                    -  High-risk CR1 including:

                         -  Poor-risk cytogenetics (e.g., Philadelphia chromosome t(9;22)or 11q23
                            rearrangements)

                         -  Presence of minimal disease by flow cytometry after 2 or more cycles of
                            chemotherapy

                    -  No CR within 4 weeks of initial treatment

                    -  Induction failure with ≤ 10% blasts in the marrow

                    -  CR2 or CR3

               -  Myelodysplastic syndromes (MDS), Intermediate, High or Very High Risk by the
                  revised international prognostic scoring system (IPSS-R)

               -  Chronic Myelogenous Leukemia (CML) in second chronic phase after accelerated or
                  blast crisis.

               -  Myelofibrosis (MF):

                    -  Intermediate-2 or high risk by Dynamic International Prognostic Scoring
                       System (DIPSS-plus) or

                    -  Monosomal karyotype or

                    -  Presence of inv(3)/i(17q) abnormalities or

                    -  Other unfavorable karyotype OR leukocytes ≥40 × 10(9) /L and

                    -  Circulating blasts ≤ 9%

               -  Relapsed or Refractory Lymphoid Malignancies (including non-Hodgkin Lymphoma,
                  Hodgkin Lymphoma and Chronic Lymphocytic Leukemia) meeting the following
                  criteria:

                    -  Disease status: Stable Disease, Partial Remission or 2nd and 3rd Complete
                       Remission. OR

                    -  Have relapsed after autologous transplant or who have failed to collect for
                       an autologous transplant.

          -  Eastern Cooperative Oncology Group (ECOG) Performance status ≤ 2

          -  Patients without a matched related or unrelated donor

          -  Patient with either one or both:

               -  Two 5/8 human leukocyte antigen (HLA) high resolution matched umbilical cord
                  blood (UCB) grafts with a cell dose of 2.0x10^7 total number of nucleated cells
                  per kilogram (TNC/kg) each, or

               -  A related haplo-identical donor

          -  Concurrent Therapy for Extramedullary Leukemia or central nervous system (CNS)
             Lymphoma: Concurrent therapy or prophylaxis for testicular leukemia, CNS leukemia, and
             CNS lymphoma including standard intrathecal chemotherapy and/or radiation therapy will
             be allowed as clinically indicated. Such treatment may continue until the planned
             course is completed. Subjects must be in CNS remission at the time of protocol
             enrollment if there is a history of CNS involvement. Maintenance therapy after
             transplant is allowed.

          -  Subjects must have the ability to understand and the willingness to sign a written
             informed consent document.

        Exclusion Criteria:

          -  Patients with inadequate Organ Function as defined by:

               -  Creatinine clearance <50ml/min

               -  Bilirubin > twice institutional upper limit of normal

               -  aspartate aminotransferase (AST) serum glutamic-oxaloacetic transaminase (SGOT) ≥
                  three times institutional upper limit of normal

               -  Alanine aminotransferase (ALT) serum glutamic pyruvic transaminase (SGPT) ≥ three
                  times institutional upper limit of normal

               -  Pulmonary function: diffusing capacity of the lung for carbon monoxide corrected
                  for hemoglobin (DLCOc) < 60% normal

               -  Cardiac: left ventricular ejection fraction < 50%

               -  Karnofsky Performance Statue (KPS) < 80

          -  Patients with uncontrolled inter-current illness including, but not limited to ongoing
             or active infection, symptomatic congestive heart failure, unstable angina pectoris,
             cardiac arrhythmia, or psychiatric illness/social situations that would limit
             compliance with study requirements.

          -  Pregnant or breastfeeding women are excluded from this study because chemotherapy
             involved with Reduced Intensity Conditioning (RIC) have the significant potential for
             teratogenic or abortifacient effects.

          -  Any condition that would, in the investigator's judgment, interfere with full
             participation in the study, including administration of study drug and attending
             required study visits; pose a significant risk to the subject; or interfere with
             interpretation of study data.

          -  Known allergies, hypersensitivity, or intolerance to any of the study medications,
             excipients, or similar compounds

          -  Presence of donor-specific antibodies against chosen graft source.