Allogeneic Hematopoietic Cell Transplantation From HLA-matched Donor After Flu-Mel-PTCy Versus Flu-Mel-ATG Reduced-intensity Conditioning

Inclusion Criteria: Patients V.1.1. Diseases Hematological malignancies confirmed histologically: * AML in morphological CR or not in morphological CR but not rapidly progressing (i.e. no need to give treatments such as hydroxyurea to maintain WBC count \< 10 000 x109/mL); * MDS; * CML in CP or AP; * MPD not in blast crisis, * MDS/MPD overlap, * ALL in CR; * Multiple myeloma; * CLL; * Non-Hodgkin's lymphoma (aggressive NHL should have chemosensitive disease); * Hodgkin's disease with chemosensitive disease or responding to checkpoint inhibitors. \* Clinical situations • Theoretical indication for a standard allo-transplant, but not feasible because: * Age \> 50 yrs; * Unacceptable end organ performance; * The physician's decision; * The patient's decision * Underlying 'lower risk' disease, for which Reduced Intensity Conditioning is preferred (eg CLL, MCL) \* Other inclusion criteria * Male or female; fertile patients must use a reliable contraception method; * Age 18-75 yrs (children of any age are not allowed in the protocol); * Informed consent given by patient or his/her guardian if indicated. Donors * Male or female; * Any age; * Human Leukocyte Antigen (HLA)-identical sibling donor or 10 of 10 (HLA-A, -B, -C, -DRB1, and -DQB1) HLA allele matched unrelated donor; * Weight \> 15 Kg (because of leukapheresis); * Fulfills criteria for allogeneic Peripheral Blood Stem Cell (PBSC) donation according to standard procedures; * Informed consent given by donor or his/her guardian if indicated, as per donor center standard procedures. Exclusion Criteria: Patients * Any condition not fulfilling inclusion criteria; * Human Immunodeficiency Virus positive; * Non-hematological malignancy(ies) (except non-melanoma skin cancer) active \< 3 years before Hematopoietic Cell Transplantation (HCT). * Life expectancy severely limited by disease other than malignancy; * Central Nervous System involvement with disease refractory to intrathecal chemotherapy. * Terminal organ failure, except for renal failure (dialysis acceptable) 1. Cardiac: Symptomatic coronary artery disease; ejection fraction \<40%; uncontrolled arrhythmia, uncontrolled hypertension; 2. Pulmonary: Diffusing Capacity of the Lung for Carbon Monoxide (DLCO)\< 40% and/or receiving supplementary continuous oxygen, Forced Expiratory Volume in 1 Second (FEV1)\< 40%; 3. Hepatic: Fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \>3 mg/dL, and symptomatic biliary disease; * Uncontrolled infection; * Karnofsky Performance Score \<70%; * Patient is a fertile man or woman who is unwilling to use contraceptive techniques during and for 12 months following treatment; * Patient is a female who is pregnant or breastfeeding; * Any condition precluding the use of melphalan or Thymoglobulin; Donors * Any condition not fulfilling inclusion criteria; * Unable to undergo leukapheresis because of poor vein access or other reasons.