This is a single arm, phase II trial of HLA-haploidentical related hematopoietic cells transplant (Haplo-HCT) using reduced intensity conditioning (fludarabine and melphalan and total body irradiation). Peripheral blood is the donor graft source. This study is designed to estimate disease-free survival (DFS) at 18 months post-transplant.

Inclusion Criteria:

• 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

Exclusion Criteria:

• Pregnant or breastfeeding
• Untreated active infection
• Active HIV infection
• Prior allogenic transplant at any time prior or less than 6 months since prior autologous transplant (if applicable)
• Active central nervous system malignancy
• Favorable risk AML defined as per protocol
• Active central nervous system malignancy
• Favorable risk AML defined as having one of the following:
• t(8,21) without cKIT mutation or evidence of immunophenotypic, cytogenetic or molecular minimal residual disease (MRD)
• inv(16) or t(16;16) without cKIT mutation or evidence of MRD
• Normal karyotype with mutated NPM1 but FLT3-ITD wild type without evidence of MRD
• Normal karyatype with double mutated CEBPA without evidence of MRD