Eligible patients will be treated with an allogeneic peripheral blood stem cell transplant from an HLA identical or single HLA antigen-mismatched family donor, using an intensive immunosuppressive regimen without myeloablation ("mini-transplant") in an attempt to decrease the transplant related toxicities while preserving the anti-malignancy and/or anti-host marrow effect of the graft. The low intensity non-myeloablative conditioning regimen should provide adequate immunosuppression to allow stem cell and lymphocyte engraftment. A T-cell replete, donor-derived, granulocyte colony stimulating factor (G-CSF)-mobilized peripheral blood stem cells (PBSC) will be used to establish hematopoietic and lymphoid reconstitution. We will infuse lymphocytes in patients with <100% donor T-cell chimerism or with evidence of tumor progression in an attempt to prevent graft rejection and enhance a graft-versus-malignancy effect, respectively.