Inclusion Criteria:

          -  Lack of a human leukocyte antigen (HLA) matched related donor, lack of an immediately
             available 8/8 HLA matched unrelated donor

          -  Patients must be diagnosed with a high-risk and/or advanced hematologic malignancy
             defined as one of the following

          -  Acute lymphocytic leukemia (ALL) in complete remission (CR)1 with high-risk features
             including adverse cytogenetic such as t(9;22), t(1;19), t(4;11), or MLL gene
             rearrangements; in second or greater morphologic remission; persistent minimal
             residual disease

          -  Acute myeloid leukemia (AML) in CR1 with intermediate-risk disease and persistent
             detectable minimal residual disease (MRD), or with high-risk features defined as:
             greater than 1 cycle of induction therapy required to achieve remission; preceding
             myelodysplastic syndrome (MDS) or myeloproliferative disease; presence of FLT3
             mutations or internal tandem duplications, DNMT3a, TET2, MLL-partial tandem
             duplication (PTD), ASXL1, PHF6; FAB M6 or M7 classification; adverse cytogenetics
             including: -5, del 5q, -7, del7q, abnormalities involving 3q, 9q, 11q, 20q, 21q, 17,
             +8, complex (> 3 abnormalities)

          -  Patients with AML must have less than 10% bone marrow blasts and < 100/mcL absolute
             peripheral blood blast count

          -  Patients with AML in CR2, subsequent CR or with active disease at transplant (< 10%
             bone marrow blasts)

          -  MDS with International Prognostic Scoring System (IPSS) intermediate-2 or higher,
             therapy-related MDS or chronic myelomonocytic leukemia (CMML)

          -  Aplastic anemia with absolute neutrophil count (ANC) < 1,000 and transfusion dependent
             after failed immunosuppression therapy

          -  Chronic myeloid leukemia (CML) >= 1st chronic phase, after failed >=2 lines of
             tyrosine kinase inhibitors; patients who progressed to blast phase must be in
             morphologic remission at transplant

          -  Relapsed Hodgkin's disease or non-Hodgkin's lymphoma (NHL)

          -  Patients with chemo-sensitive chronic lymphocytic leukemia (CLL)/small lymphocytic
             lymphoma (SLL) with persistent or recurrent disease after fludarabine-based regimens
             with < 25% involvement by CLL/SLL cells

          -  Patients with lymphoblastic lymphoma in remission or after partial response to
             chemotherapy

          -  Patients with poor prognosis multiple myeloma by cytogenetics del13, del 17p, t(4;14)
             or t(14;16) or hypodiploidy, with advanced disease (stage >= 2) and /or relapsed after
             autologous stem cell transplant

          -  Zubrod performance status 0-1 or Karnofsky performance status > 70%; patients > 50
             years will have to have a Sorror Comorbidity Index =< 3

          -  Available haploidentical donor willing and eligible to undergo a peripheral blood
             collection

          -  Left ventricular ejection fraction (LVEF) > 40%

          -  Bilirubin =< 1.5 mg/dl (unless Gilbert's syndrome), alanine aminotransferase (ALT) or
             aspartate aminotransferase (AST) =< 200 IU/ml for adults; conjugated (direct)
             bilirubin < 2 x upper limit of normal

          -  Serum creatinine clearance >= 50 ml/min (calculated with Cockcroft-Gault formula)

          -  Diffusing capacity for carbon monoxide (DLCO) >= 45% predicted corrected for
             hemoglobin

          -  Patient or patient's legal representative must provide written informed consent

        Exclusion Criteria:

          -  Human immunodeficiency virus (HIV) positive; active hepatitis B or C

          -  Patients with active infections; the principal investigator (PI) is the final arbiter
             of the eligibility

          -  Liver cirrhosis with greater than grade 1 stage 1 inflammation/fibrosis

          -  Uncontrolled central nervous system (CNS) involvement by tumor cells within the past 2
             months

          -  History of another primary malignancy that has not been in remission for at least 3
             years; (the following are exempt from the 3-year limit: nonmelanoma skin cancer, fully
             excised melanoma in situ [stage 0], curatively treated localized prostate cancer, and
             cervical carcinoma in situ on biopsy or a squamous intraepithelial lesion on PAP
             smear)

          -  Positive beta human chorionic gonadotropin (HCG) test in a woman with child bearing
             potential defined as not post-menopausal for 12 months or no previous surgical
             sterilization

          -  Inability to comply with medical therapy or follow-up