A Phase I Study of NK Cell Infusion Following Allogeneic Peripheral Blood Stem Cell Transplantation From Related or Matched Unrelated Donors in Pediatric Patients With Solid Tumors and Leukemias

* INCLUSION CRITERIA: PATIENTS (RECIPIENT) * Hematologic Malignancies Diagnoses: 1. Acute lymphoblastic leukemia (ALL) with a history of bone marrow relapse in clinical remission (CR) #2 or greater, or in CR#1 if prior induction failure; or with an M1 marrow if unable to achieve CR. 2. Philadelphia chromosome positive ALL patients who; 1. Have progressed through or relapsed following tyrosine kinase inhibitor (TKI) therapy or conventional myeloablative therapy OR 2. Are ineligible to receive tyrosine kinase inhibitor (TKI) therapy AND myeloablative hematopoietic stem cell transplant (HSCT) 3. Acute Myelogenous Leukemia (AML) with a history of bone marrow relapse in remission CR #2 or greater; or with an M1 marrow if unable to achieve CR; or in CR#1 if prior induction failure; or any of the following High-Risk categories: 1. Fms-related tyrosine kinase 3 (FLT3)/internal tandem duplication (ITD)+ with high allelic ratio \> 0.4 (high allelic ration (HR) FLT3/ITD+) regardless of low risk features. 2. Presence of monosomy 7, monosomy 5, or del5q, without inv(16)/t(16;16) or t(8;21) cytogenetics or NPM or CEBP(alpha) mutations. 3. Acute myelogenous leukemia (AML) without inv(16)/t(16;16), t(8;21), nucleophosmin (NPM), CCAAT/enhancer binding protein (CEPB)(alpha) mutations, monosomy 7, monosomy 5, del5q, or HR FLT3/ITD+, but with evidence of residual AML (greater than or equal to 0.1%) at end of Induction I. 4. Hodgkin's and Non-Hodgkin's Lymphoma with refractory disease or relapse after at least one salvage regimen, or after autologous stem cell transplant 5. Juvenile Myelocytic Leukemia (JMML) with less than 10% blasts in marrow and blood, who are not eligible for effective standard therapies. * Age: 4 to less than or equal to 35 years old at the time of enrollment for solid tumor patients and 4 to less than or equal to 35 years old for hematologic malignancies. * All previous cytotoxic chemotherapy must be completed at least 3 weeks prior to study entry. Any prior non-hematologic vital organ toxicity (cardiac, pulmonary, hepatic, renal) of any previous therapy must have resolved to grade 1 or less, unless specified elsewhere in Inclusion Criteria for Patient (Recipient). EXCEPTIONS: There is no time restriction in regard to prior intrathecal chemotherapy provided there is complete recovery from any acute toxic effects; or Subjects receiving standard acute lymphoblastic leukemia (ALL) maintenance chemotherapy will not require washout. * All previous immunologic or molecularly targeted therapy must be completed at least 1 week prior to study entry. Any prior non-hematologic toxicity of any previous therapy must have resolved to grade 1 or less, unless specified elsewhere in Inclusion Criteria for Patient (Recipient). * Prior investigational therapy must be completed at least 30 days prior to study entry * Patients with prior autologous or allogeneic transplant are eligible. Patients must be greater than 100 days post transplant and have no evidence of active graft versus host disease (GVHD). * Performance status: Eastern Cooperative Oncology Group (ECOG) 0, 1 or 2, or for children less than or equal to 10 years of age, Lansky greater than or equal to 60. Life expectancy greater than 3 months. * Availability of human leukocyte antigen (HLA)-matched (5-6/6 antigen or 8/8 allele) related or unrelated donor. * Cardiac function: Left ventricular ejection fraction greater than or equal to 45% by multi-gated acquisition scan (MUGA) or echocardiogram (ECHO), fractional shortening greater than or equal to 28% by ECHO. * Pulmonary function: Diffusing capacity of the lungs for carbon monoxide (DLCO) \>= 40% of the expected value corrected for alveolar volume and hgb for reduced intensity transplant and DLCO \>=55% for myeloablative regimen. For children who are unable to cooperate for pulmonary function tests (PFTs), the criterion is: No evidence of dyspnea at rest, no exercise intolerance, and no requirement for supplemental oxygen therapy. * Liver function: Serum total bilirubin less than 2 mg/dl, serum aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 2.5 times upper limit of normal. Patients with Gilbert syndrome are excluded from the requirement of a normal bilirubin. (Gilbert syndrome is found in 3-10% of the general population, and is characterized by mild, chronic unconjugated hyperbilirubinemia in the absence of liver disease or overt hemolysis). * Renal function: Age-adjusted normal serum creatinine according to the following, or a creatinine clearance greater than or equal to 60 ml/min/1.73 m(2): * For age (years) of less than or equal to 5, a Maximum serum creatinine (mg/dl) of 0.8 * For age (years) of greater than 5 but less than or equal to 10, a Maximum serum creatinine (mg/dl) of 1.0 * For age (years) of greater than 10 but less than or equal to 15, a Maximum serum creatinine (mg/dl) of 1.2 * For age (years) of greater than 15, a Maximum serum creatinine (mg/dl) of 1.5 * Marrow function: Absolute neutrophil count (ANC) must be greater than 750/mm(3) (unless due to underlying disease in which case there is no grade restriction), platelet count must be greater than or equal to 75,000/mm(3) (not achieved by transfusion) unless due to underlying disease in which case there is no grade restriction). Lymphopenia, cluster of differentiation 4 (CD4) lymphopenia, leukopenia, and anemia will not render patients ineligible. * Ability to give informed consent. For patients less than 18 years of age their legal guardian must give informed consent. Pediatric patients will be included in age-appropriate discussion in order to obtain verbal assent. * Durable power of attorney form completed (patients greater than or equal to 18 years of age only). * Female patients (and when relevant their male partners) must be willing to practice birth control (including abstinence) during and for two months after treatment, if of childbearing potential. EXCLUSION CRITERIA: PATIENT (RECIPIENT) * Uncontrolled infection. * Active central nervous system (CNS) malignancy as defined by: 1. Solid Tumors: History of untreated CNS tumor involvement. Extradural masses which have not invaded the brain parenchyma or parameningeal tumors without evidence for leptomeningeal spread will not render the patient ineligible. Patients with previous CNS tumor involvement are eligible IF the CNS tumor(s) has been treated and has been stable or resolving for at least 6 months; and if the patient does not currently require steroids. 2. Lymphoma: tumor mass on computed tomography (CT) scan or leptomeningeal disease 3. Leukemia: CNS 2 or CNS 3 classification. * Lactating or pregnant females (due to risk to fetus or newborn). * Human immunodeficiency virus (HIV) positive (due to unacceptable risk associated with severe immune suppression). * Hepatitis B surface antigen (HBsAg) positive or hepatitis C antibody positive with elevated liver transaminases. All patients with chronic active hepatitis (including those on treatment) are ineligible. * Patients who require systemic corticosteroid or other immunosuppressive therapy. Immunosuppressive therapy must be stopped at least 28 days prior to protocol cycle 1, day 1 (C1D1). Topical agents and/or inhaled corticosteroids are permitted. * High risk of inability to comply with transplant protocol, or inability to give appropriate informed consent in the estimation of the principal investigator (PI), social work, psychiatry, or the stem cell transplant team. * Fanconi Anemia * Clinically significant systemic illness (e.g. serious active infections or significant cardiac, pulmonary, hepatic or other organ dysfunction), that in the judgment of the PI would likely compromise the patients ability to tolerate protocol therapy or significantly increase the risk of complications. INCLUSION CRITERIA: DONOR * Weight greater than or equal to 15 kilograms and for unrelated donors, greater than or equal to 18 years. * HLA-matched related or unrelated allogeneic donors. Genotypically identical twins may serve as stem cell donors. Related donors must be 5 or 6/6 antigen matched. Unrelated donors must be 8/8 allele matched. * For donors less than 18 years of age, he/she must be the oldest suitable donor, their legal guardian must give informed consent, the donor must give verbal assent, and he/she must be cleared by social work and a mental health specialist to participate. * For donors greater than or equal to 18 years of age, ability to give informed consent. * Adequate peripheral venous access for apheresis or consent to use a temporary central venous catheter for apheresis. * Donor selection will be in accordance with National Institutes of Health (NIH)/Clinical Center (CC) Department of Transfusion Medicine (DTM) criteria and, in the case of an unrelated donor, the National Marrow Donor Program (NMDP) standards and Food and Drug Administration (FDA) 21 Code of Federal Regulations (CFR) 1271. EXCLUSION CRITERIA: DONOR * History of medical illness that in the estimation of the PI or DTM/NMDP physician poses prohibitive risk to donation including, but not limited to, stroke, hypertension that is not controlled with medication, or heart disease. Individuals with symptomatic angina or a history of coronary bypass grafting or angioplasty will not be eligible. * Anemia (Hemoglobin (Hb) less than 11 gm/dl) or thrombocytopenia (less than100,000/microliters). * Identical twins will be excluded; the lack of Major histocompatibility complex Major histocompatibility complex (MHC) incompatibility will alter the toxicity profile in such a way as to make the results uninterpretable. * Breast feeding or pregnant females. Donors of childbearing potential must use an effective method of contraception during the time they are receiving filgrastim. The effects of cytokine administration on a fetus are unknown and may be potentially harmful. The effects upon breast milk are also unknown and may potentially be harmful to the infant. * High risk of inability to comply with protocol requirements as determined by the principal investigator and donor center team. * Positive screening test for transfusion-transmissible infection in accordance with DTM or NMDP donation standards, including HIV-positive, hepatitis B surface antigen (HBsAg) positive or hepatitis C antibody positive.