Haploidentical (Half-matched) Related Donor Stem Cell Transplantation Using Killer Immunoglobulin-like Receptors in Addition to Normal Selection Factors to Determine the Best Donor

Inclusion Criteria: Patients with any of the following hematologic malignancies who are considered to be eligible for allogeneic transplantation: * Acute lymphoid leukemia (ALL) in first complete remission (CR1) with high riskfor relapse including: * t(9;22) or detected BCR-ABL1 translocation by genomic methodologies * BCR-ABL1-Like B-ALL \[54\] including mutations of IKZF1 or CRLF2 * Translocations or mutations involving 11q23 (MLL) gene. * Hypodiploid karyotype * Deletion of 9p * Loss of 17p or TP53 mutation * T-lymphocyte lineage antigen expression (T-ALL) * CNS or other extramedullary involvement * WBC count \>/= 100,000 cells/μL at diagnosis * Relapsed ALL, biphenotypic/bilineal leukemia, or AML with \</= 10% blasts in the bone marrow prior to transplantation * Acute biphenotypic or bilineal leukemia in first or greater complete remission. * Acute myeloid leukemia (AML) in CR1 with intermediate or high risk features including: * Cytogeneic abnormalities associated with myelodysplatic syndrome including abnormalities of chromosome 5 or 7 * History of anti-neoplastic therapy (radiation or chemotherapy) * Extramedullary involvement * WBC count \>/= 100,00 cells/ul at diagnosis * Rearrangements or mutations of 11q23 (MLL) * Abnormalities of chromosome 3 * TP53 mutation or loss of 17p * Complex or monosomal karyotype * Normal karyotype with mutations of FLT3, RUNX1, or ASXL1 * Myleodysplastic syndrome, myeloproliferative neoplasms, or MDS/MPN overlap syndrome with: * International prognostic scoring system risk score of INT-2 or high risk at the time of transplant evaluation * Any risk category if life-threatening cytopenia exists * Karyotype or genomic changes that indicate high risk for progression to acute myelogenous leukemia, including abnormalities of chromosome 7 or 3, mutations of TP53, or complex or monosomal karyotype * Myelofibrosis with DIPSS scores of INT-2 or high risk or any risk category if life threatening cytopenias are present * Chronic myelomonocytic leukemia (CMML) * Chronic myeloid leukemia (CML) who have failed or are intolerant to BCR-ABL tyrosine kinase inhibitors * CML with BCR-ABL mutation consistent with poor response to tyrosine kinase inhibition (e.g. T351 l mutation) * CML with accelerated or blast phase with \<20% blasts after therapy * Hodgkin lymphoma: * Relapsed disease with progression after autologous bone marrow transplant or are ineligible for this procedure * Responding to therapy prior to enrollment * Non-Hodgkin lymphoma: * Responding to therapy prior to enrollment * Progression after autologous bone marrow transplant or are ineligible for this procedure * Chronic lymphocytic leukemia with high risk disease as defined by the EBMT consensus criteria * Patients aged 18 through 69 years old are eligible * Patients aged 70-75 with HCT-CI of 0-1 are eligible * High risk hematologic malignancies * Patients must have Karnofsky performance status \>/= 70% * Cardiac left ventricular ejection fraction \>/= 50% at rest * Total bilirubin \</= 2 mg/dL, except for patients with Gilbert's syndrome * AST and ALT \</= 5x ULN unless thought to be disease related * Estimated or measured creatinine clearance \> 50 mL/min * Hemoglobin adjusted pulmonary DLCO \>/= 50% of predicted, if Hgb is within normal range, unadjusted DLCO must be \>/= 50% Exclusion Criteria: * Persons with a HLA matched sibling donor. * Female patients who are pregnant or breast-feeding * Persons with an infection that is not responding to antimicrobial therapy * Persons who are seropositive for HIV. * Persons with uncontrolled central nervous system malignancy •Persons who do not meet the age and organ function criteria specified above Presence of psychiatric or neurologic disease, or lack of social support that limits the patient's ability to comply with the treatment protocol including supportive care, follow-up, and research tests. * Prior diagnosis of non-hematologic malignancy within 5 years of planned protocol therapy EXCEPT: * Diagnosis of breast ductal carcinoma in situ treated with curative intent * Diagnosis of prostate adenocarcinoma with Gleasons score \</= 6 treated with curative intent * Non-melanomatous skin cancer