This phase I/II study the side effects and efficacy of natural killer cells after donor stem cell transplant and how they treat patients with myeloid malignancies or lymphoproliferative disorders. Investigators expanded NK cells ex vivo with a non-feeder cell regimen to avoid the risk of infusion of feeder cells with expanded NK cells. Investigators infuse NK cells after myeloablative conditioning therapy. These cells may help decrease relapse of malignant disease, severe graft versus host disease, reactivation of certain viruses, and, therefore, prolong the survival of participants.

Inclusion Criteria:

1. Patient with no matched related donor who has a related haploidentical donor identified (=< 7/8 allele match at the A, B, C, DR loci) who is willing to undergo a bone marrow harvest and an NK cell collection approximately 2 weeks of the recipient's admission for transplant; the donor must be 16 years of age or older and weigh at least 50kg.
2. Patients with one of the following diseases: acute myeloid leukemia (AML): a. first complete remission with high-risk features defined as (i) greater than 1 cycle of induction therapy required to achieve remission; (ii) preceding myelodysplastic syndrome (MDS); (iii) presence of FLT3 mutations or internal tandem duplication or other mutations associated with poor-risk AML (e.g., DNMT3A, TET2); (iv) French-American-British Classification (FAB) M6 or M7 classification; (v) adverse cytogenetics: -5, deletion (del) 5q, -7, del7q, abnormalities involving 3q, 9q, 11q, 20q, 21q, 17, +8 or complex karyotype (> 3 abnormalities); (vi) treatment-related AML, or b. second or more significant remission; patients beyond the second remission have to be in complete remission (CR) at transplant to be eligible, or c. primary induction failure with partial response to therapy who achieve adequate cytoreduction
3. Patients with myelodysplastic syndromes (MDS): a. de novo MDS with intermediate or high-risk International Prognostic Scoring System (IPSS) scores; patients with intermediate-1 features should have failed to respond to hypomethylating agent therapy or b. patients with treatment-related MDS
4. Chronic myeloid leukemia (CML): a. failed to achieve cytogenetic remission or have a cytogenetic relapse after treatment with at least 2 tyrosine kinase inhibitors, or b. accelerated phase or blast phase at any time
5. Lymphoma: a. refractory to or released from 1st line therapy, salvaged by 2nd line therapy, or b. high-risk lymphoma with CR to 1st line therapy.
6. Performance score of at least 70% by Karnofsky or 0 to 1 by Eastern Cooperative Oncology Group (ECOG) (age >= 12 years), or Lansky Play-performance scale of at least 70% or greater (age < 12 years)
7. Serum creatinine clearance equal to or more than 50 ml/min (calculated with Cockcroft-Gault formula)
8. Bilirubin equal or less than 1.5 mg/dl except for Gilbert's disease Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) equal to or less than 200 IU/ml for adults
9. Conjugated (direct) bilirubin less than 2 x upper limit of normal
10. Left ventricular ejection fraction equal to or greater than 40%
11. Diffusing capacity of the lung for carbon monoxide (DLCO) equal or greater than 50% predicted corrected for hemoglobin; for children =< 7 years of age who are unable to perform pulmonary function tests (PFT), oxygen saturation >= 92% on room air by pulse oximetry

11. Patient or patient's legal representative, parent(s), or guardian should provide written informed consent; the permission of a minor if participant's age is at least seven and less than eighteen years

Exclusion Criteria:

1. Human immunodeficiency virus (HIV) positive; active hepatitis B or C
2. Uncontrolled infections; principal investigator (PI) is the final arbiter of this criterion Liver cirrhosis
3. Central nervous system (CNS) involvement within three months
4. Positive pregnancy test in a woman with childbearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization
5. Inability to comply with medical therapy or follow-up