Gene Modified Immune Cells (IL13Ralpha2 CAR T Cells) After Conditioning Regimen for the Treatment of Stage IIIC or IV Melanoma or Metastatic Solid Tumors

Eligibility Criteria

• * Histologically confirmed malignancy that is considered surgically incurable with either:
• * Stage IIIC melanoma including locally relapsed, satellite, in-transit lesions or bulky draining node metastasis
• * Stage IV melanoma including patients with known brain metastases
• * Other metastatic, non-central nervous system (CNS) solid tumor relapsed or refractory after all standard-of-care systemic therapies for which the patient is eligible
• * Confirmed IL13Ralpha2 tumor expression by immunohistochemistry (immunohistochemical assay \[IHA\] H-Score \>= 50 in at least 10% of the total tumor specimen and in at least two high-power fields)
• * Age greater than or equal to 18 years old and less than 70 years old
• * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1
• * A minimum of one measurable lesion defined as:
• * Meeting the criteria for measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST), OR
• * Skin lesion(s) selected as non-completely biopsied target lesion(s) that can be accurately measured and recorded by color photography with a ruler to document the size of the target lesion(s)
• * Absolute neutrophil count (ANC) \>= 1 x 10\^9 cells/L (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Platelets \>= 75 x 10\^9/L (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Hemoglobin \>= 9.5 g/dL (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Aspartate and alanine aminotransferases (AST, ALT) =\< 2.5 x upper limit of normal (ULN) (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Total bilirubin =\< 2 x ULN (except patients with documented Gilbert's syndrome) (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Creatinine \< 2 mg/dL (or a glomerular filtration rate \> 45) (determined within 30-60 days prior to enrollment; re-evaluated within 14 days of beginning conditioning chemotherapy)
• * Patients with melanoma must have progressed following \>= 1 line of systemic therapy, including immune checkpoint inhibitor and a BRAF inhibitor in combination with MEK inhibitor for patients with BRAF V600-activating mutation and is not considered to have an alternate treatment option with curative intent
• * Must be willing and able to accept at least one leukapheresis procedure (This does not apply for patients receiving a second infusion of IL13R a2 CAR T cells as they will not undergo leukapheresis)
• * Must be willing and able to provide written informed consent
• * Inability to purify \>= 1 x 10\^7 T cells from leukapheresis product (this does not apply to patients receiving a second infusion of IL13Ra2 CAR T cells as they will not undergo leukapheresis)
• * Previously known hypersensitivity to any of the agents used in this study; known sensitivity to cyclophosphamide or fludarabine
• * Received systemic treatment for cancer, including immunotherapy, within 14 days prior to initiation of conditioning chemotherapy administration within this protocol
• * Clinically active brain metastases. Radiological documentation of absence of active brain metastases at screening is required for all patients. Prior evidence of brain metastasis successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment
• * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or received systemic steroids within the last 2 weeks prior to enrollment; not including patients with primary or secondary adrenal insufficiency who require physiologic replacement with steroids, or patients on inhaled or topical steroids at standard doses
• * Human immunodeficiency virus (HIV) seropositivity or other congenital or acquired immune deficiency state, which would increase the risk of opportunistic infections and other complications during chemotherapy-induced lymphodepletion. If there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist
• * Hepatitis B or C seropositivity with evidence of ongoing liver damage, which would increase the likelihood of hepatic toxicities from the chemotherapy conditioning regimen and supportive treatments. If there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist
• * Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol
• * A Tiffeneau-Pinelli index \< 70% of the predicted value. Subjects will be excluded if pulmonary function tests indicate they have insufficient pulmonary capability
• * Patients will be excluded if they have a history of clinically significant electrocardiography (ECG) abnormalities, symptoms of cardiac ischemia or arrhythmias and have a left ventricular ejection fraction (LVEF) \< 45% on a cardiac stress test (stress thallium, stress multigated acquisition scan (MUGA), dobutamine echocardiogram, or other stress test)
• * Patients with ECG results of any conduction delays (PR interval \> 200 ms, corrected QT (QTC) \> 480 ms), sinus bradycardia (resting heart rate \< 50 beats per minute), sinus tachycardia (HR \> 120 beats per minute) will be evaluated by a cardiologist prior to starting the trial. Patients with any arrhythmias, including atrial fibrillation/atrial flutter, excessive ectopy (defined as \> 20 ventricular premature complex \[PVC\]s per minute), ventricular tachycardia, 3rd degree heart block will be excluded from the study unless cleared by a cardiologist
• * Pregnancy or breast-feeding. Female patients must be surgically sterile or be postmenopausal for two years, or must agree to use effective contraception during the period of treatment and for 6 months afterwards. All female patients with reproductive potential must have a negative pregnancy test (serum/urine) at screening and again within 14 days from starting the conditioning chemotherapy. The definition of effective contraception will be based on the judgment of the study investigators. Patients who are breastfeeding are not allowed on this study
• * A concomitant active malignancy that would be considered to interfere with the assessment of the primary or secondary endpoints of the study