Mesenchymal Stem Cell Infusion as Prevention for Graft Rejection and Graft-versus-host Disease

V.1. Patients V.1.1. Diseases Hematological malignancies confirmed histologically and not rapidly progressing: * AML in CR; * ALL in CR; * CML unresponsive/intolerant to Imatinib but not in blast crisis; * Other myeloproliferative disorders not in blast crisis and not with extensive myelofibrosis; * MDS with \< 5% blasts; * Multiple myeloma; * CLL; * Non-Hodgkin's lymphoma (aggressive NHL should have chemosensitive disease); * Hodgkin's disease. V.1.2. Clinical situations * Theoretical indication for a standard allo-transplant, but not feasible because: * Age \> 55 yrs; * Unacceptable end organ performance; * Patient's refusal. * Indication for a standard auto-transplant: perform mini-allotransplantation 2-6 months after standard autotransplant. V.1.3. Other inclusion criteria * Male or female; fertile female patients must use a reliable contraception method; * Age \< 75 yrs. * Informed consent given by patient or his/her guardian if of minor age. V.1.4. Exclusion criteria * Any condition not fulfilling inclusion criteria; * HIV positive; * Terminal organ failure, except for renal failure (dialysis acceptable); * Uncontrolled infection, arrhythmia or hypertension; * Previous radiation therapy precluding the use of 2 Gy TBI; * HLA-identical donor. V.2. PBSC donors V.2.1. Inclusion criteria * Related to the recipient (sibling, parent or child) or unrelated; * Male or female; * Weight \> 15 Kg (because of leukapheresis); * Fulfills generally accepted criteria for allogeneic PBSC donation; * Informed consent given by donor or his/her guardian if of minor age, as per donor center standard procedures. V.2.2. Exclusion criteria * Any condition not fulfilling inclusion criteria; * HIV positive; * Unable to undergo leukapheresis because of poor vein access or other reasons. V.2.3. HLA matching Related or unrelated donors who have 1-2 HLA mismatches, as either : * One antigenic mismatch at HLA-A or -B or -C or -DRB1 or -DQB1 * One allelic mismatch at HLA-A or -B or -C or -DRB1 or -DQB1 * Two allelic mismatches at HLA-A or -B or -C or -DRB1 or -DQB1 * One antigenic mismatch + 1 allelic mismatch at HLA-A or -B or -C or -DRB1 or -DQB1. * One antigenic mismatch at -DQB1 and one other antigenic mismatch at HLA-A or -B or -C or -DRB1 V.3. Cord blood unit Banked cord blood units will be used if they fulfill the following criteria: * No more than 2/6 HLA mismatches (antigenic mismatch at HLA-A or HLA-B or allelic mismatch at HLA-DRB1) * \> 2.5 x 107 TNC/kg * Standard validation by FACT/Netcord criteria.