Inclusion Criteria:  Recipient Inclusion Criteria  Patients with the following diseases that are histopathologically-confirmed are eligible  Acute myeloid, lymphoid, or mixed phenotype leukemia in complete remission (CR) or CR with incomplete hematologic recovery (CRi) or beyond first complete remission (CR1) without the presence of minimal residual disease Acute myeloid, leukemia, or mixed phenotype leukemia that is either: Not in morphologic CR with bone marrow infiltration by leukemic blasts of ≤10%, or In morphologic CR with evidence of minimal residual disease positivity by either multiparametric flow cytometric analysis or by a nucleic acid-based technique Primary refractory acute myeloid, lymphoid, or mixed phenotype leukemia Chronic myelogenous leukemia (accelerated, blast or second chronic phase) Myelodysplastic syndromes Myelofibrosis that is transplant-eligible Myeloproliferative syndromes Blastic plasmacytoid dendritic cell neoplasm Non-Hodgkin lymphoma with poor risk features not suitable for autologous HCT Match to the patient as follows:  a. For Arm A:  Availability of a 8/8 or 7/8 HLA-matched donor (related or unrelated) defined by Class I (HLA-A, -B, -C) serologic typing (or higher resolution) and Class II (HLA-DRB1) molecular typing. If the donor is a 7/8 HLA-match, the mismatch must be a permissive allelic mismatch as assessed by an independent HLA and transplantation expert. b. For Arm B: Availability of a haploidentical donor who is a ≥ 4/8 but <7/8 match at HLA-A, -B, -C, and -DRB1 (typed using DNA-based high-resolution methods), with at most one mismatch per locus c. Age ≥ 18 and ≤75 years old at the time of enrollment. d. Left ventricular ejection fraction (LVEF) ≥ 45% e. Diffusing capacity of the lungs for carbon monoxide (DLCO) ≥ 50% f. Calculated creatinine clearance ≥ 50 mL/min or creatinine < 2.0 mg/dL g. SGPT and SGOT ≤ 5 x ULN, unless elevated secondary to disease Total bilirubin ≤ 3 x ULN (patients with Gilbert's syndrome may be included at the discretion of the PI or where hemolysis has been excluded h. Negative serum or urine beta-HCG test in females of childbearing potential within 3 weeks of registration i. Karnofsky performance status ≥ 70%  Donor Inclusion Criteria  Age ≥ 18 and ≤ 75 years of age Karnofsky performance status of ≥ 70% defined by institutional standards Seronegative for HIV-1 RNA PCR; HIV 1 and HIV 2 ab (antibody); HTLV-1 and HTLV-2 ab; PCR+ or sAg (surface antigen) hepatitis B ; or PCR or sAg negative for hepatitis C; negative for the Treponema palladum antibody Syphillis screen; and negative for HIV-1 and hepatitis C by nucleic acid testing (NAT) within 30 days of apheresis collection. In the case that T palladum antibody tests are positive, donors must:  Be evaluated and show no evidence of syphilis infection of any stage by physical exam and history Have completed effective antibiotic therapy to treat syphilis Have a documented negative non-treponemal test (such as RPR) or in the case of a positive non-treponemal test must be evaluated by an infectious disease expert to evaluate for alternative causes of test positivity and confirm no evidence of active syphilitic disease e. Match to the patient as follows:  a. Arm A:  Must be a related or unrelated, 8/8 or 7/8-HLA match to recipient at HLAA, -B, -C, and -DRB1. If 7/8 HLA-matched, must be with permissive allelic HLA mismatch as assessed by an independent HLA and transplantation expert. b. Arm B:  Must be a haploidentical donor who is ≥ 4/8 but < 7/8 match at HLA-A, -B,  C, and -DRB1, with at most one mismatch per locus. f. Must be willing to donate PBSC for up two consecutive days g. Female donors of child-bearing potential must have a negative serum or urine beta HCG test within 3 weeks of mobilization h. Capable of undergoing leukapheresis, have adequate venous access, and be willing to undergo insertion of a central catheter should leukapheresis via peripheral vein be inadequate i. Agreeable to 2nd donation of PBPC (or bone marrow harvest) in the event of graft failure j. The donor or legal guardian greater than 18 years of age, capable of signing an IRB approved consent form. k. Meets other criteria for donation as specified by standard NMDP guidelines (NMDP donors) or institutional standards (non-NMDP donors)  Exclusion Criteria:  Recipient Exclusion Criteria  Seropositive for any of the following:  HIV antibodies; hepatitis B surface antigen (sAg); hepatitis C antibodies  Prior myeloablative therapy or hematopoietic cell transplant Patients deemed candidates for fully myeloablative preparative conditioning regimens Candidate for autologous transplant HIV-positive Active uncontrolled bacterial, viral or fungal infection, defined as currently taking antimicrobial therapy and progression of clinical symptoms. Uncontrolled CNS disease involvement Pregnant or a lactating female Positive serum or urine beta-HCG test in females of childbearing potential within 3 weeks of registration Psychosocial circumstances that preclude the patient being able to go through transplant or participate responsibly in follow-up care Known allergy or hypersensitivity to, or intolerance of, tacrolimus Hematopoietic cell transplantation-specific comorbidity index (HCT-CI) ≥ 5  Positive anti-donor HLA antibodies against a mismatched allele in the selected donor determined by either:  A positive crossmatch of any titer; or The presence of anti-donor HLA antibody to any HLA locus Any uncontrolled autoimmune disease requiring active immunosuppressive treatment Concurrent malignancies or active disease within 1 year, except nonmelanomatous skin cancers that have been curatively resected  Donor Exclusion Criteria  Evidence of active infection Seropositive for HIV-1 or-2, HTLV-1 or -2 Medical, physical, or psychological reason that would place the donor at increased risk for complications from growth factor or leukapheresis Lactating female