Inclusion Criteria:

1. Acute Myelogenous Leukemia (AML), Myelodysplastic Syndrome (MDS): induction failure, high-risk for relapse first remission (with high-risk cytogenetics, flt3 mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from MDS, Langerhan's cell histiocytosis, second or third complete remission, or second relapse.
2. Acute Lymphoblastic Leukemia (ALL): induction failure, first complete remission with Philadelphia chromosome or translocation (4;11), hypodiploidy, and/or evidence of minimal residual disease by flow cytometry;second third complete remission, or second relapse.
3. CML second chronic phase or accelerated phase.
4. Non-Hodgkin's Lymphoma (NHL): Induction failures, second or third complete remission, or relapse (including relapse post autologous)
5. Hodgkin's Disease (HD): Induction failures, second or third complete remission,or relapse (including relapse post autologous hematopoietic stem cell transplant).
6. Chronic Lymphocytic Leukemia (CLL): Progressive disease following standard therapy
7. Multiple Myeloma: stage II or III , symptomatic , secretory Multiple Myeloma requiring treatment.
8. Age greater than or equal to 1 year but less than or equal to 55 years (Myeloablative Regimen 4). Eligibility for pediatric patients will be determined in conjunction with an MDACC pediatrician. Patients >55 but < 65 years who have a Performance Status of 0 or 1 and no comorbidities may receive the myeloablative regimen 4 at the discretion of the investigator(s).
9. Age greater than 55 years and less than or equal to 80 years (Nonmyeloablative Regimen 2)
10. Age greater than or equal to 1 but less than or equal to 80 years old that in the opinion of the investigator(s) would preclude myeloablative therapy and who cannot receive Total Body Irradiation (TBI) may receive reduced intensity regimen 3.
11. Performance score of at least 60% by Karnofsky or PS less than 3 (ECOG) (age greater than or equal to 12 years), or Lansky Play-Performance Scale of at least 60% or greater (age <12 years)
12. Left ventricular ejection fraction of at least 40% (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or 30% (Nonmyeloablative Regimen 2)
13. Pulmonary function test demonstrating a diffusion capacity of least 50% predicted (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or at least 40% predicted (Nonmyeloablative Regimen 2). For children < 7 years of age who are unable to perform PFT, oxygen saturation > 92% on room air by pulse oximetry.
14. Creatinine < 1.6 mg/dL (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or < 3.0 mg/dL (Nonmyeloablative Regimen 2).
15. SGPT/bilirubin < / = to 2.0 x normal (Myeloablative Regimen 4, Reduced Intensity Regimen 3) or < / = 4.0 x normal (Nonmyeloablative Regimen 2)
16. Negative Beta HCG test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on study.
17. Unrelated Cord Blood will be used as a source of hematopoietic support if a 5 or 6/6 related or 6/6 unrelated bone marrow donor is not available, or if the tempo of a patient's disease dictates it is not in the patient's best interest to wait for an unrelated marrow donor to be procured.
18. Patients must have two Cord Blood units available which are matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 10 million total nucleated cells/Kg recipient body weight (pre-thaw)
19. Patients must have a family member who is matched at 2, 3, or 4 HLA antigens typed as described above and willing to donate 80-100 ml or bone marrow for MSC generation or the Angioblast Mesenchymal Precursor Cells will be used for the cord blood co-cultures. Patients that are high risk for relapse are eligible to use the Angioblast "off-the-shelf" Mesenchymal Precursor Cells.
20. Patient must be willing to undergo bone marrow harvest or peripheral blood progenitor cells collection for use in case of engraftment failure.
21. If the patient is unable / fails to successfully undergo autologous bone marrow harvest or peripheral blood progenitor cells collection a family member must be identified who agrees to donate bone marrow or peripheral blood progenitor cells for T-depleted transplant. A third cord blood unit could be used in case of engraftment failure. This cord blood unit must be identified prior to enrollment in this study.
22. If the family member is to be used as the donor, he/she must freely document his / her willingness to comply with this donation by signing a donor informed consent form prior to preparative regimen for transplantation. This donor could be the same person who donates the bone marrow aspirate for mesenchymal stem cell generation

Exclusion Criteria:

1. HIV positive
2. Positive Pregnancy Test
3. Uncontrolled serious medical condition such as persistent septicemia despite adequate antibiotic therapy, decompensated congestive heart failure despite cardiac medications or pulmonary insufficiency requiring intubation. (excluding primary disease for which CB transplantation is proposed), or psychiatric condition that would limit informed consent.
4. Patients with active (untreated) CNS disease
5. Availability of appropriate, willing, HLA-matched related marrow donor.