Brief Summary: This is a phase I, open-label, non-randomized study that will enroll pediatric and young adult research participants with relapsed or refractory non-CNS solid tumors to evaluate the safety, feasibility, and efficacy of administering T cell products derived from the research participant's blood that have been genetically modified to express a B7H3-specific receptor (chimeric antigen receptor, or CAR) that will target and kill solid tumors that express B7H3. On Arm A of the study, research participants will receive B7H3-specific CAR T cells only. On Arm B of the study, research participants will receive CAR T cells directed at B7H3 and CD19, a marker on the surface of B lymphocytes, following the hypothesis that CD19+ B cells serving in their normal role as antigen presenting cells to T cells will promote the expansion and persistence of the CAR T cells. Arm A CAR T cells include the protein EGFRt and Arm B CAR T cells include the protein HER2tG. These proteins can be used to both track and destroy the CAR T cells in case of undue toxicity. The primary objectives of the study will be to determine the feasibility of manufacturing the cell products, the safety of the T cell product infusion, to determine the maximum tolerated dose of the CAR T cells products, to describe the full toxicity profile of each product, and determine the persistence of the modified cell in the participant's body on each arm. Participants will receive a single dose of T cells comprised of two different subtypes of T cells (CD4 and CD8 T cells) felt to benefit one another once administered to the research participants for improved potential therapeutic effect. The secondary objectives of this protocol are to study the number of modified cells in the patients and the duration they continue to be at detectable levels. The investigators will also quantitate anti-tumor efficacy on each arm. Participants who experience significant and potentially life-threatening toxicities (other than clinically manageable toxicities related to T cells working, called cytokine release syndrome) will receive infusions of cetuximab (an antibody commercially available that targets EGFRt) or trastuzumab (an antibody commercially available that targets HER2tG) to assess the ability of the EGFRt on the T cells to be an effective suicide mechanism for the elimination of the transferred T cell products.

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