Evaluating the Safety and Effectiveness of an Umbilical Cord Blood Stem Cell Transplant (BMT CTN 0604)

Inclusion Criteria:

* Participants must be 21 to 70 years old; participants 1 to 21 years old are also eligible if they are ineligible for BMT CTN #0501 (NCT00412360)
* Each unit must supply a minimum of 1.5 x 10\^7/kg pre-cryopreserved nucleated cell dose
* Participants must have two partially human leucocyte antigen (HLA)-matched umbilical cord blood units. Each unit must match at a minimum of 4 of 6 at HLA-A, -B, -DRB1 loci with the recipient. This may include 0 to 2 antigen mismatches at each A or B (at the antigen level) or DRB1 (at the allele level) loci. Each unit must be a 4 to 6 HLA-A, B, and DRB1 antigen matched to each other, not necessarily at the same loci as with the recipient. All typing will be done using molecular typing. Though molecular level typing will be available, a match is defined at intermediate resolution for HLA-A and -B and at high resolution for -DRB1 for this study. An adult unrelated donor search is not required for a person to be eligible for this study if the clinical situation dictates an urgent transplant. Clinical urgency is defined as 6 to 8 weeks from referral to transplant center or low likelihood of finding a matched, unrelated donor.
* Must have received cytotoxic chemotherapy within 3 months of the consent date (measured from the start date of chemotherapy)
* Acute leukemias (includes T lymphoblastic lymphoma) in the second or subsequent complete remission (CR)
* Burkitt's lymphoma in the second or subsequent CR
* Lymphoma
* Patients with adequate physical function, as measured by the following:

  * Heart: left ventricular ejection fraction at rest greater than 35%, or shortening fraction greater than 25%
  * Liver: bilirubin less than or equal to 2.5 mg/dL and alanine aminotransferase (ALT), aspartate aminotransferase (AST), and alkaline phosphatase less than or equal to five times the upper limit of normal
  * Kidney: serum creatinine within normal range for age, or if serum creatinine is outside the normal range for age, then kidney function (creatinine clearance or glomerular filtration rate (GFR) greater than 40 mL/min/1.73m\^2
  * Lungs: forced expiratory volume in one second (FEV1), forced vital capacity (FVC), and carbon monoxide diffusing capacity (DLCO) greater than 50% predicted (corrected for hemoglobin). If unable to perform pulmonary function tests, then oxygen (O2) saturation must be greater than 92% on room air.

Exclusion Criteria:

* Have an HLA-matched, related, or 7 or 8/8 HLA allele matched (HLA-A, -B, -Cw, -DRB1) related donor able to donate
* Had an autologous hematopoietic stem cell transplant in the 3 months before study entry
* Pregnant or breastfeeding
* Evidence of HIV infection or known HIV positive serology
* Current uncontrolled bacterial, viral, or fungal infection (i.e., currently taking medication with evidence of progression of clinical symptoms or radiologic findings)
* Prior allogeneic hematopoietic stem cell transplant
* History of primary idiopathic myelofibrosis