Inclusion 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

Exclusion Criteria:

   - 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