OBJECTIVES: I. Determine the rates of hematologic and immune reconstitution in pediatric patients with high risk hematologic malignancies in first remission or in second or subsequent remission, and adult patients with acute lymphocytic leukemia (ALL) or acute nonlymphocytic leukemia (ANLL) in second or subsequent remission, who are undergoing high dose chemoradiotherapy followed by unrelated umbilical cord blood (UCB) transplantation. II. Determine the incidence of graft-versus-host disease in this setting. III. Determine whether contamination of umbilical cord blood with maternal cells is a clinical problem in this setting. IV. Describe the incidence of leukemic relapse in these patients after UCB transplantation. V. Describe the incidence of serious infections and secondary lymphoproliferative diseases following transplantation with UCB in these patients. VI. Determine specifically whether larger recipients (greater than 40 kg) can be durably engrafted with unrelated UCB, and determine whether nucleated cell or progenitor cell content of the graft is predictive of hematological engraftment.