Donor Stem Cell Transplant in Treating Patients With Previously Treated Lymphoma, Multiple Myeloma, or Chronic Lymphocytic Leukemia

Inclusion Criteria: * Donors will be \<55 years of age and in good health as approved by the National Marrow Donor Program (NMDP) donor and collection centers. Related donors will be \< 70 years of age. * Recipients will be \<55 years, will have satisfactory organ function (excluding bone marrow) and will have a Karnofsky activity assessment \>90% and will have: * Creatinine \<2.0 mg/dl. * Bilirubin, Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<2 x normal. * Pulmonary function test and Carbon Monoxide Diffusing Capacity (DLCO) \> 50% of normal. * Multi Gated Acquisition Scan (MUGA) \>45% injection fraction. * Recipients with unrelated donor matched at the HLA A, B, DRBI loci, or if \< 35 years mismatched at a single HLA A or B, or DRBI locus. * Umbilical cord blood (5) used as an unrelated stem cell source will provide \> 2.0 x 10\^7 cells/kg and will be matched at 4 - 6 of 6 HLA A, B, and DRBI loci. Cord blood grafts may include a single or pair of cord units depending on the cell dose. * Partially matched related donors will be at least haploidentical (matched at \>3 of 6 HLA A, B, DRB1 loci). * Recipients will fall under one of the following disease categories * Chronic lymphocytic leukemia -- must have all three: * Rai Stage III/IV * Progression after previous Complete Response (CR) or Partial Response (PR) including purine antagonist (i.e. fludarabine). * Recent chemotherapy responsiveness * Advanced non-Hodgkin's lymphoma(NHL). * Low-grade NHL (Working Formulation A, B, C) following progression after initial therapy if asymptomatic at diagnosis (\>CR2, \>PR2; response duration \< 1 year from last therapy) or if no CR was achieved (\>PR1). At least one prior therapy of intermediate intensity (e.g. CHOP). * Mantle zone lymphoma after any progression following initial therapy (\>CR1, \> PR1). At least one prior therapy of intermediate intensity (e.g. CHOP). * Intermediate grade lymphoma (\>PR2). Response duration \<1 year from prior therapy. * High-grade Non-Hodgkin's Lymphoma (NHL) (IWF H, I, J) after initial therapy if \>stage III at diagnosis; after any progression even if localized (stage I, II) at diagnosis with prior response duration \< 1 year. * Recent chemotherapy responsiveness after treatment with \> 3 intermediate intensity regimens. * Advanced Hodgkin's disease beyond PR2 (\>CR3, \>PR3). * Recent chemotherapy responsiveness * Multiple Myeloma (\>CR2, \>PR2) or after initial therapy if no prior PR. * Recent chemotherapy responsiveness * Recipients will sign informed consent approved by the Committee on the Use of Human Subjects at the University of Minnesota. Exclusion Criteria: * No available histocompatible related donor; 2nd bone marrow transplant (BMT), HIV-1 positive; active uncontrolled infection; or resistant malignancy.