Reduced Intensity Allogeneic HCT in Advanced Hematologic Malignancies w/T-Cell Depleted Graft

This study is testing a type of bone marrow transplant (called allogeneic HCT) that uses a less intense preparation regimen and a specially processed donor graft to reduce side effects, for people with advanced blood cancers like leukemia or myelodysplastic syndrome. **You may be eligible if...** - You have been diagnosed with acute myeloid or lymphoid leukemia, chronic myelogenous leukemia, myelodysplastic syndrome, or a related blood cancer - Your disease is in remission, or in certain cases, has a small amount remaining (minimal residual disease) - A matched or partially matched bone marrow donor is available for you - You are in adequate overall health to undergo a transplant **You may NOT be eligible if...** - You have active uncontrolled infection - Your organ function (heart, liver, lungs, kidneys) is significantly impaired - You have already had a prior allogeneic transplant (depending on study arm) - You are pregnant or breastfeeding Talk to your doctor to see if this trial is right for you.