This protocol provides expanded access to bone marrow transplants for children who lack a histocompatible (tissue matched) stem cell or bone marrow donor when an alternative donor (unrelated donor or half-matched related donor) is available to donate. In this procedure, some of the blood forming cells (the stem cells) are collected from the blood of a partially human leukocyte antigen (HLA) matched (haploidentical) donor and are transplanted into the patient (the recipient) after administration of a "conditioning regimen". A conditioning regimen consists of chemotherapy and sometimes radiation to the entire body (total body irradiation, or TBI), which is meant to destroy the cancer cells and suppress the recipient's immune system to allow the transplanted cells to take (grow). A major problem after a transplant from an alternative donor is increased risk of Graft-versus-Host Disease (GVHD), which occurs when donor T cells (white blood cells that are involved with the body's immune response) attack other tissues or organs like the skin, liver and intestines of the transplant recipient. In this study, stem cells that are obtained from a partially-matched donor will be highly purified using the investigational CliniMACS® stem cell selection device in an effort to achieve specific T cell target values. The primary aim of the study is to help improve overall survival with haploidentical stem cell transplant in a high risk patient population by limiting the complication of GVHD.