This study if for patients that have a blood disease and it's been determined that the best option for treating that blood disease is a cord blood transplant. Cord blood (CB) is blood that is taken from the umbilical cord and placenta of healthy newborn babies after childbirth. The cord blood collected from a newborn baby is called a cord blood unit. The United States Food and Drug Administration (FDA) considers cord blood to be a biological drug. These are considered "investigational" products. This study will evaluate the safety of administration of the investigational cord blood units by carefully documenting all infusion-related problems.

The primary purposes of this study are to:
•Provide access to cord blood units for recipients whose best choice for a cord blood unit(s) do not meet all FDA standards, but do meet standards set by the NMDP on this study.
•Assess how well and how quickly blood counts return to normal after transplant in recipients on this study.