Inclusion Criteria:

1. Patients with multiple myeloma (MM), acute myeloid leukemia (AML), myelodysplastic syndrome (MDS), acute lymphoblastic leukemia (ALL), non-Hodgkin's lymphoma (NHL), Hodgkin's lymphoma (HL), or chronic lymphocytic leukemia (CLL) in greater than first complete remission who are candidates for a non-ablative or reduced intensity conditioning regimen.
2. Age up to 80 years.
3. A related or unrelated donor who is HLA-matched at HLA, A, B, C, DR and DQ loci is acceptable (i.e. 10/10 matched related or unrelated donor, matched with molecular high-resolution technique per current standard for the BMT program). Donor must be willing to donate peripheral blood or bone marrow progenitor cells.
4. Available cord blood unit must contain a minimum of 1.5 x 10^7 total nucleated cells per kg, and be at least a 4/6 HLA match with patient.
5. Zubrod PS less than or equal to 2 or Lansky PS greater than or equal to 50%.
6. Left ventricular ejection fraction >40%. No uncontrolled arrhythmias or symptomatic heart disease.
7. Forced Expiratory Volume in 1 second (FEV1), Forced Vital Capacity (FVC) and Diffusion Capacity (DLCO) >40%.
8. Serum creatinine <2.0 mg/dL. Serum bilirubin <3 X upper limit of normal, SGPT <4 X upper limit of normal.

Exclusion Criteria:

1. Patients with active CNS disease
2. Positive Beta HCG in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization.
3. Serious medical or psychiatric illness likely to interfere with participation in this clinical study, including but not limited to active uncontrolled infection, uncontrolled cardiac arrhythmia or ischemic event, or uncontrolled psychosis, major depression, or mania.
4. Evidence of chronic, active hepatitis or cirrhosis, or HIV