After hospital discharge, patients are followed in the clinic at least once or twice weekly for a medical history, physical exam, and blood tests for 100 days. They receive medications to prevent infection and GVHD and, if needed, blood transfusions. If GVHD has not developed by about 120 days post transplant, patients receive additional white cells to boost the immune response. After 100 days, follow-up visits may be less frequent. Follow-up continues for at least 5 years. During the course of the study, patients undergo repeated medical evaluations, including blood tests and radiology studies, to check on the cancer and on any treatment side effects. On four occasions, white blood cells may be collected through apheresis to see if immune responses can be generated against the sarcomas treated in this study. Positron emission tomography (PET) scans may be done on five occasions. This test uses a radioactive material to produce images useful in detecting primary tumors and cancer that has spread.