In this study we will evaluate the use of T cell depleted peripheral blood progenitor cells (PBPC) (instead of bone marrow) to optimize the stem cell and lymphocyte dose. Donors will be given G-CSF and their mobilized PBPC harvested by leukapheresis. To minimize acute GVHD, the transplant will be T cell depleted, using a new technique developed in normal volunteers which improves T cell depletion and reduces stem cell loss (protocol 96-H-0049). The study has two phases: The first phase evaluates engraftment and GVHD following T cell depleted PBPC transplants. Stopping rules will be used to make modifications to the protocol in the event of graft failure. Cyclosporine will be withdrawn from the protocol if the incidence of acute GVHD is low or absent. In the second phase patients will receive add-back of donor lymphocytes on day 45 and day 100 post transplant to prevent relapse and confer donor-immune function. The risk of acute GVHD following this procedure will be determined. It is planned to treat up to 55 patients aged between 10 and 60 years. The end points of the study are graft take; acute and chronic GVHD, leukemic relapse, transplant-related and all causes of mortality, cytomegalovirus reactivation and leukemia-free survival. Patients will be followed for 5 years.