This study hopes to show that specially treated umbilical cord cells, called stem cells, can be safely given to a person after they receive chemoradiation therapy or chemotherapy for their illness. During chemoradiation therapy or chemotherapy, a person loses all of the cells that are needed to make the different types of cells in their blood, including their immune system cells. These cells must be replaced in order for the blood and immune systems to work properly. Some people receive bone marrow transplants or other types of stem cell transplants to get the cells they need. CB001 is being developed as an option for people who need bone marrow transplants or other types of transplants to replace those cells. It is also being developed for people who do not have the option of other types of transplants.

Inclusion Criteria:

• Do not have identical or 5/6 related matched bone marrow, peripheral blood or umbilical cord
• Stable disease and lack of unrelated donor
• Acute myeloid leukemia (AML) in 2nd or subsequent complete remission or first remission with high risk features
• ALL in 2nd or subsequent remission or first remission with high risk features
• Myelodysplastic syndrome (MDS)
• Non-Hodgkin Lymphoma (NHL)
• Chronic Myelogenous Leukemia (CML)
• Adequate function of heart, liver, kidneys and lungs

Exclusion Criteria:

• Females who are pregnant
• Poor ability to perform daily activities
• Weight under 40 kilograms (88 pounds)
• AML caused by chemoradiation
• Prior stem cell transplant
• Uncontrolled infection at time of transplant
• Active fungal infection
• HIV infection
• Primary myelofibrosis
• Receiving other research drugs
• Unable to provide informed consent