GPC3 targeted CAR-T cell therapy for advanced liver cancer

* INCLUSION CRITERIA:
* Histopathological confirmation of HCC or other solid tumor malignancy by the NCI Laboratory of Pathology
* Participants must:

  * have progressed on at least 1 prior line of treatment

OR

--been intolerant of at least 1 prior line of treatment.

* Participants must have at least 1 focus of disease that is amenable to mandatory tumor biopsy prior to study treatment initiation to determine tumor GPC3 expression and be willing to undergo this. Ideally, the biopsied lesion should not be one of the target measurable lesions, although this can be up to the discretion of the investigators.
* Tumor must have GPC3 positivity of \>= 25% by immunohistochemistry on freshly collected biopsy
* Participants must have at least 1 measurable lesion by RECIST version 1.1
* Participants must have a disease that is not amenable to potentially curative resection, ablation, or transplantation.
* Age \>= 18 years.
* Performance status (ECOG) 0-1
* Participants must have adequate organ and marrow function as defined below:

ANC: \>= 1,000/mcL

Platelets: \>= 75,000/mcL

Hemoglobin: \>= 8 g/dL

total bilirubin: If cirrhosis present: Part of Child Pugh requirement

If no cirrhosis: bilirubin should be \<= 1.5 x ULN

ALT or AST: \<= 5 x ULN.

Creatinine OR Measured or calculated creatinine clearance (CrCl) (eGFR may also be used in place of CrCl) (A): \< 1.5x institution upper limit of normal OR \>= 50 mL/min/1.73 m\^2 for participant with creatinine levels, \>= 1.5 X institutional ULN

ALT (SGPT)=alanine aminotransferase (serum glutamic pyruvic transaminase);

AST (SGOT)=aspartate aminotransferase (serum glutamic oxaloacetic transaminase); GFR=glomerular filtration rate; ULN=upper limit of normal.

(A)Creatinine clearance (CrCl) or eGFR should be calculated per institutional standard.

* Normal cardiac ejection fraction (\>= 50% by echocardiogram) and no evidence of hemodynamically significant pericardial effusion as determined by an echocardiogram within 4 weeks before treatment initiation.
* Room air oxygen saturation of 92% or greater.
* Treatment-related toxicities must be resolved to \<= grade 1.
* For participants with brain metastases: Participants with \<=3 (three or fewer) brain metastases that have been treated with surgery or stereotactic radiosurgery or other form of treatment are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month before protocol treatment.
* The study drugs are harmful to developing human fetus. For this reason, women of childbearing potential must agree to use highly effective contraception (hormonal, intrauterine device (IUD), abstinence, surgical sterilization) at the study entry and up to 12 months after the last dose of combined chemotherapy. Men must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) at the study entry and up to 4 months after the last dose of study drugs. We also recommend men with partners of childbearing potential ask their partners to be on highly effective birth control (hormonal, IUD, surgical sterilization). Men must not freeze or donate sperm within the same period.
* HBV infected participants must be on antivirals and have HBV DNA \< 100IU/mL. HCV infected participants can be enrolled with close HCV RNA level monitoring.
* Participants must be able to understand and be willing to sign a written informed consent.
* For participants that do not have a legally authorized representative in place, one must be identified before study treatment starts

Exclusion Criteria

* Prior systemic therapy, an investigational therapy, radiation, and/or surgery within 2 weeks prior to treatment initiation.
* Prior administration of anti-PD-1 or anti-PD-L1 antibodies or other agents that in the opinion of the PI can stimulate immune activity and interfere with an infusion of CAR-T cells within 8 weeks prior to treatment initiation.
* Child-Pugh class B or C liver function
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements.

Note: Participants with a history of abnormal pulmonary function tests but stable obstructive or restrictive pulmonary disease may be eligible per PI discretion.

* Any form of primary immunodeficiency (e.g. severe combined immunodeficiency).
* HIV-positive participants are excluded because HIV causes complicated immune deficiency and study treatment can pose more risks for these participants.
* Participants receiving systemic steroids \>= 0.5 mg prednisone equivalent/kg/day. Steroid creams, ointments, and eye drops are allowed. Dose adjustment or discontinuation of medication must occur at least 24 hours prior to conditioning chemotherapy. Use of CART cell therapy in autoimmune diseases has the potential to be associated with serious safety risk. Given that this is an evolving area of research, caution should be exercised and any decision to include participants with autoimmune diseases should be made on a case-bycase basis.
* History of severe immediate hypersensitivity reaction to cyclophosphamide or fludarabine.
* Hospitalization within 7 days prior to treatment initiation.
* Pregnant women are excluded from this study because study therapy can cause fetal harm. Because there is a potential risk for adverse events in nursing infants secondary to treatment of the mother with study therapy, breastfeeding should be discontinued if the mother is treated with study drugs.
* Participants who received live or attenuated vaccine or virus-based vaccine within 30 days before initiation of study therapy
* Participants with a history of seizure disorder
* Participants with an expected life expectancy of less than 3 months before initiation of study therapy.