If patients are found to not be in remission at screening, then the patient may be returned to their primary hematologist/oncologist or may receive chemotherapy as per standard of care for the malignant disease. Patients for whom this would be their first allogeneic transplant must be in remission (< 5% malignant blasts in marrow and peripheral blood and no evidence of extramedullary disease) for transplant. Patients enrolled on this protocol for their second transplant do not need to have attained remission prior to transplant.