Baricitinib for the Prophylaxis of Graft-Versus-Host Disease After Peripheral Blood Hematopoietic Cell Transplantation

Inclusion Criteria: Patients must meet the following criteria within 30 days prior to Day 0 unless otherwise noted. * Diagnosis of a hematological malignancy listed below: * Acute myelogenous leukemia (AML) in complete morphological remission (based on IWG Criteria). * Acute lymphocytic leukemia (ALL) in complete morphological remission (MRD negative, based on IWG Criteria). * Myelodysplastic syndrome with less than 10% blasts in bone marrow. * Non-Hodgkin's lymphoma (NHL) or Hodgkin's disease (HD) in 2nd or greater complete or partial remission. * Planned treatment is myeloablative or reduced intensity conditioning followed by peripheral blood HLA matched donor transplantation * Available HLA-identical donor who meets the following criteria: * At least 18 years of age. * HLA-identical donor/recipient match by high-resolution typing per institutional standards. * In the investigator's opinion, is in general good health, and medically able to tolerate leukapheresis required for harvesting HSC. * No active hepatitis. * Negative for HTLV and HIV. * Not pregnant. * Donor selection will be in compliance with institutional standards * Safety Lead-In Phase: For the first three patients at each dose level, related donors must consent to a second product collection should it prove necessary. * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Adequate organ function as defined below: * Total bilirubin must be within normal range at baseline. * AST (SGOT) and ALT (SGPT) ≤ 3.0 x IULN. * Estimated creatinine clearance ≥ 60 mL/min/1.73 m2 by Cockcroft-Gault Formula. * Oxygen saturation ≥ 90% on room air. * LVEF ≥ 40%. * FEV1 and FVC ≥ 40% predicted, DLCOc ≥ 40% predicted. If DLCO is \< 40%, patients will still be considered eligible if deemed safe after a pulmonary evaluation. * At least 18 years of age at the time of study registration * Able to understand and willing to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). * Must be able to receive GVHD prophylaxis with tacrolimus, mini-methotrexate with or without ATG or post transplant Cy with MMF and tacrolimus as outlined in the protocol Exclusion Criteria: * Must not have undergone a prior allogeneic donor (related, unrelated, or cord) transplant. Prior autologous transplant is not exclusionary. * Known HIV or active hepatitis B or C infection. * Known latent tuberculosis infection, or at high risk for latent TB infection, or a positive t-spot tuberculosis test * Known hypersensitivity to one or more of the study agents, including baricitinib. * Must not have myelofibrosis or other disease known to prolong neutrophil engraftment to \> 35 days after transplant. * Currently receiving or has received any investigational drugs within the 14 days prior to the first dose of study drug (Day -3). * Pregnant and/or breastfeeding. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, autoimmune disease, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmias, or psychiatric illness/social situations that would limit compliance with study requirements. * Immunosuppressive doses of steroids. Subjects with steroids for adrenal insufficiency will not be excluded. * History of unprovoked thrombosis or known thrombophilia. Provoked and/or superficial DVTs are eligible provided they are treated and resolved at the time of screening. * Recent (less than 1 year from screening) myocardial infarction or embolic stroke