CD19/CD22 Chimeric Antigen Receptor (CAR) T Cells in Children and Young Adults With Recurrent or Refractory CD19/CD22-expressing B Cell Malignancies

* ELIGIBILITY CRITERIA: Study subjects with cluster of differentiation 19 (CD19) + cluster of differentiation 22 (CD22+) expressing B cell malignancies who have relapsed or are treatment refractory may enroll as defined by the following inclusion and exclusion criteria. INCLUSION CRITERIA: * Diagnosis * Participant must have a B cell acute lymphoblastic leukemia (ALL) (inclusive of chronic myeloid leukemia (CML) with ALL transformation) or lymphoma and must have relapsed or refractory disease after at least one standard chemotherapy regimen and one salvage regimen. In view of the principal investigator (PI) and the primary oncologist, there must be no available alternative curative therapies and subjects must be either ineligible for allogeneic stem cell transplant (SCT), have refused SCT, recurred after SCT, or have disease activity that prohibits SCT at the time of enrollment. Participants who have undergone autologous SCT will be eligible, and participants that have undergone allogeneic SCT will be eligible if, in addition to meeting other eligibility criteria, they have no evidence of graft versus host disease (GVHD) and have been without immunosuppressive agents for at least 30 days. Participants with Philadelphia chromosome + ALL must have failed prior tyrosine kinase inhibitor. * Participants must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis. For those being considered for reinfusions, measurable or evaluable disease is not required at the time of reinfusion. * CD22/CD19 expression --CD19 expression must be detected on greater than 15% of the malignant cells by immunohistochemistry or greater than 90% by flow cytometry. The choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each participant. In general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples. CD22+ B cell malignancy is required and CD22 expression levels will be documented when available, but a specific level of expression is not an eligibility requirement; it may be documented as positive or negative. * Age: --Greater than or equal to 3 years of age (and at least 15 kg) and less than or equal to 39 years of age at time of enrollment (greater than or equal to 3 years to less than or equal to 39 years). NOTE: The first participant in each dose cohort must be greater than or equal to 18 years of age. * Clinical Performance --Clinical performance status: Participants greater than or equal to 16 years of age: Karnofsky greater than or equal to 50%; Participants \< 16 years of age: Lansky scale greater than or equal to 50%. Subjects who are unable to walk because of paralysis, but who are upright in a wheelchair will be considered ambulatory for the purpose of calculating the performance score. * Participants must have adequate organ and marrow function as defined below: * leukocytes greater than or equal to 750/mcL\* * platelets greater than or equal to 50,000/mcL\* * total bilirubin less than or equal to 2 X upper limit of normal (ULN) (except in the case of subjects with documented Gilbert's disease \> 3x ULN) * Aspartate aminotransferase (AST)serum glutamic-oxaloacetic transaminase (SGOT)/alanine aminotransferase (ALT)serum glutamate pyruvate transaminase (SGPT) less than or equal to 10 X institutional upper limit of normal * creatinine less than or equal to the maximum for age listed in the table below * Age (Years): less than or equal to 5. Maximum Serum Creatinine (mg/dL): less than or equal to 0.8 * Age (Years): 6 to less than or equal to 10. Maximum Serum Creatinine (mg/dL): less than or equal to 1.0 * Age (Years): \>10. Maximum Serum Creatinine (mg/dL): less than or equal to 1.2 * OR * creatinine clearance greater than or equal to 60 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal. * if these cytopenias are not judged by the investigator to be due to underlying disease (i.e. potentially reversible with anti-neoplastic therapy); A subject will not be excluded because of pancytopenia greater than or equal to Grade 3 if it is due to disease, based on the results of bone marrow studies. * Subjects with central nervous system (CNS) disease are eligible, with exceptions as noted in the exclusion criteria * Contraception: --Individuals of child-bearing or child-fathering potential (IOCBP or IOCFP) must be willing to practice birth control from the time of enrollment on this study and for 12 months after receiving the preparative regimen for IOCBP and for 4 months after receiving the preparative regimen for IOCFP * Cardiac function: Left ventricular ejection fraction greater than or equal to 45% or fractional shortening greater than or equal to 28%, and no clinically significant electrocardiogram (ECG) findings * Pulmonary Function * Baseline oxygen saturation \>92% on room air at rest * Participants with respiratory symptoms must have a diffusing capacity for carbon monoxide (DLCO)/adjusted \> 45%. For children who are unable to cooperate for pulmonary function tests (PFTs) they must not have dyspnea at rest or known requirement for supplemental oxygen. * Ability of subject, parent(s)/guardian(s), Legally Authorized Representative (LAR) or Durable Power of Attorney (DPA) to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: Subjects meeting any of the following criteria are not eligible for participation in the study: * Recurrent or refractory ALL limited to isolated testicular. * Subjects with radiologically detected active CNS lymphoma or isolated CNS disease which are eligible for definitive CNS directed radiation therapy will be excluded. * Hyperleukocytosis (greater than or equal to 50,000 blasts/micro L) or rapidly progressive disease that in the estimation of the investigator and sponsor would compromise ability to complete study therapy; * Pregnant or nursing individuals. * Subjects will be excluded related to the following prior therapy criteria: * Systemic chemotherapy, anti-neoplastic investigational agents, or antibody-based therapies =\<2 weeks (6 weeks for clofarabine or nitrosoureas) prior to apheresis with the following exception: ---No time restriction with prior intrathecal chemotherapy, steroid therapy, hydroxyurea or ALL maintenance type chemotherapy (vincristine, 6-mercaptopurine, oral methotrexate, or a tyrosine kinase inhibitor for participants with Philadelphia chromosome (Ph)+ ALL) provided there is recovery from any acute toxic effects. * Radiation therapy =\<3 weeks prior to apheresis with the following exception: ---No time restriction with radiation therapy if the volume of bone marrow treated is less than 10% and the subject has measurable/evaluable disease outside the radiation window. * History of allogeneic stem cell transplantation prior to apheresis that meet the following criteria: * Less than 100 days post-transplant * Evidence of active graft-versus-host disease (GVHD) * Taking immunosuppressive agents within 30 days prior to apheresis * Less than 6 weeks post donor lymphocyte infusion (DLI) * History of prior CAR therapy or other adoptive cell therapies prior to apheresis that meet the following criteria: * Less than 30 days post-infusion * Circulating CAR T cells (or genetically modified cells) \>5% by flow cytometry in peripheral blood. * Human Immunodeficiency Virus (HIV)/hepatitis B virus (HBV)/hepatitis C virus (HCV) Infection: * a. Seropositive for HIV antibody. (Participants with HIV are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy in the future should study results indicate effectiveness.) * b. Positive for Hepatitis B surface antigen (HbsAG). * c. Evidence of active Hepatitis C (evidenced by detectable HCV ribonucleic acid (RNA) * Uncontrolled, symptomatic, intercurrent illness including but not limited to infection, congestive heart failure, unstable angina pectoris, cardiac arrhythmia, psychiatric illness, or social situations that would limit compliance with study requirements or in the opinion of the PI would pose an unacceptable risk to the subject; * Second malignancy other than in situ carcinoma of the cervix, unless the tumor was treated with curative intent at least two years previously and subject is in remission; * History of severe, immediate hypersensitivity reaction attributed to compounds of similar chemical or biologic composition to any agents used in study or in the manufacturing of the cells.