Reduced-Intensity Fludarabine, Melphalan, and Total Body Irradiation Conditioning for Transplantation of HLA-Haploidentical Related Hematopoietic Cells (Haplo-HCT) For Patients With Hematologic Malignancies

• Age ≥ 55 years or HCT Co-Morbidity score (HCT-CI) \>/=3

• Lack of a suitable 8/8 HLA-matched sibling donor

• Adequate performance status is defined as Karnofsky score ≥ 70%

• Patients and selected donor must be HLA typed at high resolution using DNA based typing at the following HLA-loci: HLA-A, -B, -C and DRB1. Donors must be HLA-haploidentical relatives including, but not limited to, children, siblings, or parents, defined as having a shared HLA haplotype between donor and patient at HLA-A, -B, -C, and -DRB1.

• Acute Myeloid Leukemia (AML): Must be in remission with morphology (\<5% blasts)

• Acute Lymphoblastic Leukemia (ALL)/lymphoma second or greater complete remission (CR) first CR unable to tolerate consolidation chemotherapy due to chemotherapy-related toxicities, first CR high-risk ALL

• Biphenotypic/Undifferentiated/Prolymphocytic Leukemias in first or subsequent CR

• Myelodysplastic syndrome: any subtype including refractory anemia (RA) if severe pancytopenia or complex cytogenetics. Blasts must be less than 5%. If 5% of more requires chemotherapy for cytoreduction to \</=5% prior to transplantation.

• Chronic Myelogenous leukemia in accelerated phase: patient must have failed at least two different Tyrosine Kinase Inhibitor (TKI)s, been intolerant to all TKIs, or have T315l mutation

• Myeloproliferative neoplasms/myelofibrosis: Blasts must be less than 5%. If 5% or more requires chemotherapy for cytoreduction to \</=5% prior to transplantation

• Relapsed large-cell lymphoma, mantle-cell lymphoma or Hodgkin lymphoma that is chemotherapy sensitive and has failed or ineligible for an autologous transplant

• Burkitt's lymphoma in second CR or subsequent CR

• Relapsed T-cell lymphoma that is chemotherapy sensitive in CR/Partial Response (PR) that has failed or ineligible for an autologous transplant

• Natural killer cell malignancies

• Relapsed chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, follicular lymphoma with any of the following:

• Progressed within 12 months of achieving a partial or complete remission Patients who had remissions lasting up

• Patients who had remission lasting \> 12 months are eligible after at least two prior therapies

• Patients with primary, refractory disease. Bulky disease and an estimated tumor doubling time of less than one month require debulking therapy prior to transplant.

• Lymphoplasmacytic lymphoma is eligible after initial therapy if chemotherapy sensitive

• Adequate organ function as defined per protocol

• Sexually active females of child bearing potential and males with partners of child bearing potential must agree to use adequate birth control during study treatment