Double Cord Blood Transplant for Patients With Malignant and Non-malignant Disorders

Inclusion Criteria: * Patients will be eligible for double cord blood stem cell transplant (TNC ≥ 4x107/kg of two combined units) if available single cord blood has TNC ≤4.0 x 107/kg and they lack a matched (5-6/6) family donor, a 10/10 unrelated adult donor, and/or if their disease status required emergent stem cell transplant and they could not wait 2-3 months for searching for a matched unrelated adult donor. * Adequate renal function defined as:Serum creatinine \<1.5 x normal, or Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>60 ml/min/m2 or \>60 ml/min/1.73 m2 or an equivalent GFR as determined by the institutional normal range. * Adequate liver function defined as:Total bilirubin \<1.5 x normal, or serum glutamic-oxaloacetic transaminase (SGOT) (aspartate aminotransferase (AST)) or serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase (ALT)) \<3.0 x normal * Adequate cardiac function defined as:Shortening fraction \>27% by echocardiogram, or Ejection fraction \>47% by radionucleotide angiogram or echocardiogram. * Adequate pulmonary function defined as:Uncorrected diffusing capacity of the lungs for carbon monoxide (DLCO) 50% by pulmonary function test.For children who are uncooperative, no evidence of dyspnea at rest, no exercise intolerance, and a pulse oximetry \>94% on room air. Eligibility for Moderate Intensity, Reduced Intensity Regimen and Fanconi's Anemia (Regimens C, D and E) * Adequate renal function defined as: Serum creatinine \<2.0 x normal, or Creatinine clearance or radioisotope GFR 40 ml/min/m2 or \>40 ml/min/1.73 m2 or an equivalent GFR as determined by the institutional normal range. * Adequate liver function defined as:Total bilirubin \<2.5 x normal, or SGOT (AST) or SGPT (ALT) \<5.0 x normal * Adequate cardiac function defined as:Shortening fraction of \>25% by echocardiogram, or Ejection fraction \>40% by radionucleotide angiogram or echocardiogram. * Adequate pulmonary function defined as:Uncorrected DLCO \>35% by pulmonary function test. For children who are uncooperative, no evidence of dyspnea at rest, no exercise intolerance, and a pulse oximetry \>94% on room air. Exclusion Criteria: * Females who are pregnant or breast-feeding * Patients with documented uncontrolled infection at the time of study entry