A dose of T cell depleted immature hematopoietic cells, wherein said T cell depleted immature hematopoietic cells comprise less than 5 x 10 5 CD3 + cells per kilogram body weight of a subject, and wherein said dose comprises at least about 5 x 10 6 CD34+ cells per kilogram body weight of said subject, and wherein said T cell depleted immature hematopoietic cells are obtained by separating said T cells from said immature hematopoietic cells, by a method selected from the group consisting of: (I) a method comprising: (i) adding an antibody to the immature hematopoietic cells that specifically binds to a surface marker, wherein said surface marker comprises a T cell surface marker selected from the group consisting of CD2, CD3, CD4, CD8 and TCRα/β, or an immature hematopoietic cell surface marker selected from the group consisting of CD34, CD33 and CD131, wherein said antibody is labeled with a magnetically responsive agent; (ii) immobilizing said immature hematopoietic cells specifically bound to said antibody labeled with said magnetically responsive agent in a matrix through a magnetic field; (iii) washing said matrix to remove unbound cells; and (iv) removing said magnetic field to elute bound cells from said matrix; and (II) based on an expression of at least one surface marker of T cells selected from the group consisting of CD2, CD3, CD4, CD8 and TCRα/β; and a therapeutically effective amount of cyclophosphamide, wherein said therapeutically effective amount comprises 25-200 mg per kilogram body weight of the subject, for use in treating a human subject in need of a non-syngeneic cell or tissue graft, and wherein said cyclophosphamide is administered to the subject following transplantation of said cell or tissue graft, and further wherein said subject is not treated with GVHD prophylaxis for more than 10 days post-transplant. (I) a method comprising: (i) adding an antibody to the immature hematopoietic cells that specifically binds to a surface marker, wherein said surface marker comprises a T cell surface marker selected from the group consisting of CD2, CD3, CD4, CD8 and TCRα/β, or an immature hematopoietic cell surface marker selected from the group consisting of CD34, CD33 and CD131, wherein said antibody is labeled with a magnetically responsive agent; (ii) immobilizing said immature hematopoietic cells specifically bound to said antibody labeled with said magnetically responsive agent in a matrix through a magnetic field; (iii) washing said matrix to remove unbound cells; and (iv) removing said magnetic field to elute bound cells from said matrix; and (i) adding an antibody to the immature hematopoietic cells that specifically binds to a surface marker, wherein said surface marker comprises a T cell surface marker selected from the group consisting of CD2, CD3, CD4, CD8 and TCRα/β, or an immature hematopoietic cell surface marker selected from the group consisting of CD34, CD33 and CD131, wherein said antibody is labeled with a magnetically responsive agent; (ii) immobilizing said immature hematopoietic cells specifically bound to said antibody labeled with said magnetically responsive agent in a matrix through a magnetic field; (iii) washing said matrix to remove unbound cells; and (iv) removing said magnetic field to elute bound cells from said matrix; and (II) based on an expression of at least one surface marker of T cells selected from the group consisting of CD2, CD3, CD4, CD8 and TCRα/β; and a therapeutically effective amount of cyclophosphamide, wherein said therapeutically effective amount comprises 25-200 mg per kilogram body weight of the subject, for use in treating a human subject in need of a non-syngeneic cell or tissue graft, and wherein said cyclophosphamide is administered to the subject following transplantation of said cell or tissue graft, and further wherein said subject is not treated with GVHD prophylaxis for more than 10 days post-transplant. <EOS>