The traditional way of doing a donor transplant is to give high doses of chemotherapy and radiation before giving the stem cells. However, high doses of chemotherapy and radiation can have serious side-effects. The doctors think that the transplant will be safer and more likely to be successful with reduced doses of chemotherapy and radiation. The purpose of this study is to find out how good a combination of chemotherapy and radiation at reduced doses followed by a cord blood transplant are at treating cancer.

The stem cells chosen for the transplant are from umbilical cord blood. Umbilical cord blood is collected from healthy newborn babies and frozen. One cord blood collection is called a "cord blood unit." On transplant day, the cord blood will be given through the catheter just like a blood transfusion. Transplants done this way have been successful. However, this type of transplant is fairly new. Therefore, it is important to study it so the doctors can better understand how it works.

Most blood or bone marrow transplants using donor stem cells are done as part of a study. When patients are on a study we test new ways of treating them which we think may be better than the old ways. We collect information about the result of this treatment so we can understand how well the treatment works. This is so we can learn better ways to treat our patients.

Inclusion Criteria:

• At least one cycle of induction or re-induction chemotherapy or lymphoma chemotherapy or azacitidine or decitabine or tyrosine kinase inhibitor.
• Patients aged 18-70 years at initial referral with no available and suitably matched related or unrelated donor.
• Acute myelogenous leukemia (AML):
• Complete first remission (CR1) at high risk for relapse such as:
• Known prior diagnosis of myelodysplasia (MDS) or myeloproliferative disorder;
• Therapy related AML;
• White cell count at presentation > 100,000;
• Presence of extramedullary leukemia at diagnosis;
• Any unfavorable sub type by FAB or WHO classification;
• High-risk cytogenetics (eg those associated with MDS, abnormalities of 5, 7, 8, Philadelphia chromosome, complex karyotype)or high risk molecular abnormalities;
• Requirement for 2 or more inductions to achieve CR1.
• Any patient with newly diagnosed AML with intermediate risk cytogenetics.
• Any patient unable to tolerate consolidation chemotherapy as would have been deemed appropriate by the treating physician.
• Complete second remission (CR2).
• Other acute leukemias that are ambiguous lineage or of other types eg blastic plasmacytoid dendritic cell neoplasm in CR1 or CR2
• Acute lymphoblastic leukemia (ALL):
• lymphoblastic leukemia (ALL):
• Complete first remission (CR1) at high risk for relapse such as:
• White cell count at presentation > 30,000 for B-cell lineage and >100,000 for Tcell lineage;
• Presence of a high-risk cytogenetic abnormality 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 after four weeks of induction therapy;
• Any patient with newly diagnosed ALL > or = to 50 years-old;
• Any patient unable to tolerate consolidation and/or maintenance chemotherapy as would have been deemed appropriate by the treating physician.
• Complete second remission (CR2).
• Myelodysplastic Syndrome (MDS):
• Intermediate-1 International Prognostic Scoring System (IPSS) score with poor risk cytogenetics as defined by IPSS.
• Intermediate- 2 or High International Prognostic Scoring System (IPSS) score.
• MDS/ myeloproliferative disorder overlap syndromes.
• Any score with life threatening cytopenia(s), including red cell or platelet transfusion dependence.
• Receipt of at least one cycle of cytotoxic chemotherapy, azacitidine or decitabine.
• MDS patients must have < or = to 5% bone marrow myeloblasts and ANC > or = to 0.2 (growth factor supported if necessary) at transplant work-up.
• Myeloproliferative Disorder (MPD)
• Life-threatening cytopenia(s), and/or red blood cell or platelet transfusion dependence
• Patients with aplasia
• Patients with excess blasts less than or equal to 10% blasts in the bone marrow at work-up.
• Chronic myelogenous leukemia (CML) patients who have failed or are intolerant of tyrosine kinase inhibitors or patients with other myeloproliferative diseases who are high risk and the intent of therapy is cure.
• Any Non-Hodgkins lymphoma (including chronic lymphocytic leukemia)or Hodgkin's lymphoma at high-risk of relapse
• Eligible patients with DLC NHL will:

have relapsed disease following initial therapy but failed to mobilize or had bone marrow involvement and therefore are not suitable for an autologous transplant OR

• have failed an autologous transplant and be in CR after salvage chemotherapy.
• Eligible patients with transformed indolent NHL/CLL will:

have CR/PR of the large cell component of their disease after either salvage chemotherapy or an autologous transplant.

• Eligible patients with mantle cell NHL will:

be high-risk as such as p53 positivity and be in 1st CR/PR after initial therapy OR have relapsed disease following initial therapy and be in 2nd or 3rd CR/PR after salvage chemotherapy.

• Eligible patients with indolent B cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) or CLL will have 2nd or subsequent progression (pre-allograft cytoreduction necessary but CR/PR not required).
• Eligible patients with HL will be without progression of disease (POD) after salvage chemotherapy.
• Timing of UCBT:
• Admission for UCBT must be within an acceptable time period after the pre-allograft chemotherapy.
• Organ Function and Performance Status Criteria:
• Karnofsky score > or = to 70 %.
• calculated creatinine clearance > or = to 60 ml/min
• bilirubin < than or = to 1.5 mg/dL, ALT < than or = to 3 x upper limit of normal unless benign congenital hyperbilirubinemia,
• pulmonary function (spirometry and corrected DLCO) > or = to 50% predicted.
• left ventricular ejection fraction > or = to 50%.
• albumin > or = to 3.0.
• Graft Criteria:
• 2 UCB units selected according to current MSKCC unit selection algorithm. High resolution 8 allele HLA typing will be performed. Unit selection will occur based on 8 allele HLA-match and CD34+ dose.
• In addition, each unit will have a cryopreserved dose of at least 1.5 x 10^7 total nucleated cells/recipient body weight (TNC/kg).
• Units with attached segments for confirmatory typing will be given preference.

Exclusion Criteria:

• Diagnosis: acute leukemia in morphologic relapse or with morphologic persistent disease (cytogenetic or molecular persistence/relapse in morphologic CR are eligible); MDS or CML or other myeloproliferative disorder with > 5% blasts; Aggressive lymphoma or HL with POD after salvage chemotherapy.
• Two prior stem cell transplants of any kind.
• One prior autologous stem cell transplant within the preceding 12 months.
• One prior allogeneic stem cell transplant within the preceding 24 months.
• Prior radiation therapy with 400cGy or more of TBI.
• Active and uncontrolled infection at time of transplantation.
• HIV infection.
• Seropositivity for HTLV-1.
• Inadequate performance status/ organ function.
• Pregnancy or breast feeding.
• Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, follow-up, and research tests