NK Cells in Cord Blood Transplantation

Inclusion Criteria: * Patients must have one of the following hematologic malignancies: acute myelogenous leukemia (AML), induction failure, high-risk for relapse first remission (with intermediate-risk or high-risk cytogenetics, fms-related tyrosine kinase 3 \[FLT3\] mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from myelodysplastic syndrome, any disease beyond first remission * Myelodysplastic syndrome (MDS): primary or therapy related * Acute lymphoblastic leukemia (ALL): induction failure, primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease; patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: with t(9;22) or t(4;11), hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure, and/or evidence of minimal residual disease, or acute biphenotypic leukemia, or double hit non-Hodgkin's lymphoma * Non-Hodgkin's lymphoma (NHL) in second or third complete remission, refractory NHL, or relapsed NHL (including relapse post autologous hematopoietic stem cell transplant); double hit lymphomas in first remission or more advanced disease * Small lymphocytic lymphoma (SLL), or chronic lymphocytic leukemia (CLL) with progressive disease following standard th…