A Study of Codrituzumab in Children and Young Adults With Solid Tumors and Have Not Responded to Treatment or Have Come Back After Treatment

Inclusion Criteria: * Consent/Assent: All patients and/or their parents or legally authorized representatives must sign written informed consent; assent, when appropriate, will be obtained according to institutional guidelines. * Age: Patients must be aged 12 months up to but not including 22 years at the time of study registration. A patient who is 21 at the time of enrollment but turns 22 thereafter will still be considered eligible for the purposes of this study. Diagnosis: * Patients must have a diagnosis of a primary extra-cranial solid tumor that is recurrent or refractory to standard therapy. * For the purposes of this study, the following definitions will be used: * Refractory is defined as any tumor which progresses despite maximal standard therapies * Recurrent (relapsed) is defined as a completion of planned therapy after which point the tumor recurs within 5 years of treatment. Additionally, any tumor which recurs twice is considered relapsed. * Tumor GPC3 Expression: Patients must have demonstrated positive GPC3 expression via immunohistochemistry (IHC) on any prior tumor sample. Confirmation of GPC3 expression may include a diagnostic or relapsed sample, at a primary or metastatic site. There is no limit to how long ago this sample was collected. This GPC3 expression via IHC will be centrally confirmed by Ventana as in prior studies of codrituzumab and described in the treatment. Patients may choose to enroll on the prescreening portion, which allows for assessment of GPC3 expression only, prior to enrollment on the full clinical trial. * Tumor GPC3 Expression: Patients must have demonstrated a minimum of 1+ GPC3 expression via immunohistochemistry (IHC) on any prior tumor sample. Confirmation of GPC3 expression may include a diagnostic or relapsed sample, at a primary or metastatic site. There is no limit to how long ago this sample was collected. This GPC3 expression via IHC will be centrally confirmed by Ventana as in prior studies of codrituzumab and described in the treatment plan section. Patients may choose to enroll on the prescreening portion, which allows for assessment of GPC3 expression only, prior to enrollment on the full clinical trial. * Disease Status: Patients must have measurable disease based on RECIST 1.1. * Performance Level: Patients must have Karnofsky Performance Score (for patients \> 16 years of age) or Lansky Performance Score (for patients ≤ 16 years of age) ≥ 50% assessed within 2 weeks of study enrollment. * Neurological Deficits: Patients with neurologic deficits must have been stable and off of steroids for a minimum of 1 week prior to study entry; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Pregnancy/Contraception: Patients must not be pregnant or breast-feeding; females, excluding pre-menstrual, must have a negative serum or urine pregnancy test within 7 days prior to enrollment; males or females of reproductive potential may not participate unless they have agreed to use an effective contraceptive method, which includes abstinence, for 90 days after the last dose of study drug. * Prior Therapy: Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study; recovery is defined as all AEs, attributable to prior therapy, having improved to grade 2 or better or as outlined below. For agents that have known adverse events occurring beyond 28 days after administration, this period must be extended beyond the time during which adverse events are known to occur; the duration of this interval must be discussed with the study chair. * Systemic Anticancer Therapy (e.g. Chemotherapy): * Not Myelosuppressive: \> 7 days must have elapsed from their last dose of anticancer therapy not known to be myelosuppressive (e.g. not associated with reduced platelet or ANC counts). * Myelosuppressive: \> 14 days must have elapsed from their last dose of known myelosuppressive anticancer therapy. * Antibodies: \> 21 days must have elapsed from the infusion of last anticancer antibody. * Cellular Therapies: \> 42 days must have elapsed from the completion of any type of cellular therapy, including modified T cells, NK cells, dendritic cells, etc. * Radiation: Patients who have had radiation must have had their last fraction of: * Local irradiation to the primary tumors or other limited sites (cumulative dose \< 40Gy) \> 14 days prior to registration. * Local irradiation to the primary tumors or other sites (cumulative dose ≥ 40Gy), therapeutic 131I-MIBG or other radiopharmaceutical, and other substantial bone marrow radiation, \> 42 days prior to registration. * Craniospinal irradiation, radiation to \> 50% of pelvis, or total body irradiation \> 120 days prior to registration. * Stem Cell Infusions: With or without TBI * 84 days must have elapsed from an allogeneic bone marrow or stem cell transplant, or any stem cell infusion including DLI or boost infusion and patients must also not have any evidence of acute or chronic GvHD. * 42 days must have elapsed from an autologous stem cell infusion including boost infusion. * Supportive Therapies: Patients must be off all growth factor(s) that support platelet, red blood cell, or white blood cell count, number or function for at least 7 days prior to registration (e.g. filgrastim, sargramostim, erythropoietin, romiplostim, eltrombopag, etc.) as well as off Pegylated G-CSF for at least 14 days prior to registration. * Organ Function: Patients must have documented within 14 days of registration and within 7 days of starting treatment the following: * Hgb \> 8 gm/dL (may be transfusion-supported) * Platelet count \> 50,000/mm\^3 (transfusion independent) * Absolute neutrophil count (ANC) \> 1000/mm\^3 * INR ≤ 2.5 * Total Bilirubin (sum of conjugated + unconjugated) ≤ 3 times institutional upper limit of normal (ULN) for age * Aspartate aminotransferase (AST) ≤ 5 times institutional ULN for age * Alanine Aminotransferase (ALT) ≤ 5 times institutional ULN for age * Serum albumin ≥ 2 g/dL * GFR ≥ 50 mL/min/1.73 m\^2 as measured using urine creatinine clearance, serum cystatin c, radioisotope GFR, or serum creatinine as measured by the Schwartz equation (Refer to Schwartz, et al. J Am Soc Neph, 2009) Exclusion Criteria: * Patients receiving current anti-cancer therapy or investigational agents are not eligible for study entry. * Patients who do not have tumor tissue available for GPC3 testing are not eligible for study entry. * Patients who have received any prior GPC3-directed immunotherapy are not eligible for study entry. * Patients with uncontrolled seizures are not eligible for study entry. * Subjects with a condition requiring systemic treatment with either corticosteroids (\>0.15 mg/kg daily prednisone equivalents) or other immunosuppressive medications, if used to modify immune adverse events related to prior therapy, \> 14 days must have elapsed since last dose of corticosteroid or immunosuppressive agent. Inhaled or topical steroids, and adrenal replacement doses ≤ 0.15 mg/kg daily prednisone equivalents are permitted in the absence of active autoimmune disease. * Patients with documented CNS tumor, CNS metastasis, CNS ischemia and/or infarction, whether symptomatic or discovered incidentally without clinical symptoms, will be excluded from study participation. * Patients with a baseline QTc \> 480. (as measured using Bazett formula; Refer to Bazett, Heart, 1920). * Patients with an inability to return for follow-up visits, obtain follow-up studies required to assess toxicity to therapy, or comply with the safety monitoring requirements. * Patients who have an uncontrolled infection are not eligible. * Patients who have received a prior solid organ transplantation are not eligible.