Inclusion Criteria:

          -  Stage IV or locally advanced unresectable cancers for which no alternative therapies
             with proven survival advantage are available

          -  NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commercially
             available NY-ESO-1 antibodies

          -  HLA-A*0201 (HLA-A2.1) positivity by molecular subtyping

          -  Age greater than or equal to 16 years old; if patients 16-17 years old are enrolled in
             the trial, they will only be enrolled after 3 patients >= 18 years old have been
             treated, and the treatment has been shown to be safe

          -  A minimum of one measurable lesion defined as:

               -  Meeting the criteria for measurable disease according to Response Evaluation
                  Criteria in Solid Tumors (RECIST)

               -  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)

          -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1

          -  Adequate bone marrow and major organ function to undergo a PBSC transplant determined
             within 30-60 days prior to enrollment using standard phase 1 criteria for organ
             function defined as:

               -  Absolute neutrophil count (ANC) >= 1.5 x 10^9 cells/L

               -  Platelets >= 100 x 10^9/L

               -  Hemoglobin >= 9 g/dL

               -  Aspartate and alanine aminotransferases (AST, ALT) =< 2.5 x ULN (upper limit of
                  normal) (=< 5 x ULN, if documented liver metastases are present)

               -  Total bilirubin =< 2 x ULN (except patients with documented Gilbert?s syndrome)

               -  Creatinine < 2 mg/dl (or a glomerular filtration rate > 60)

          -  Must be willing and able to accept at least three leukapheresis procedures

          -  Must be willing and able to undergo three research PET scans

          -  Must be willing and able to provide written informed consent

        Exclusion Criteria:

          -  Inability to purify >= 2.5 x 10^6 CD34-enriched cells/kg of patient weight from the
             pooled G-CSF mobilized leukapheresis products

          -  Previously known hypersensitivity to any of the agents used in this study; known
             sensitivity to busulfan or fludarabine

          -  Received systemic treatment for cancer, including immunotherapy, within 28 days prior
             to initiation of conditioning chemotherapy administration within this protocol

          -  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 (inhaled or topical steroids at standard doses are allowed)

          -  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

          -  Known clinically active brain metastases; 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
             and there are no neurological signs of potential brain metastases

          -  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) within 14 days from
             starting the conditioning chemotherapy; the definition of effective contraception will
             be based on the judgment of the study investigators

          -  Since IL-2 is administered following cell infusion:

               -  Patients will be excluded if they have a history of clinically significant
                  electrocardiogram (ECG) abnormalities, symptoms of cardiac ischemia with evidence
                  of ischemia on a cardiac stress test (stress thallium, stress multigated
                  acquisition [MUGA], dobutamine echocardiogram or other stress test)

               -  Similarly, patients with a baseline left ventricular ejection fraction (LVEF) <
                  45% will be excluded.

               -  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 (heart rate >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 premature ventricular contractions [PVCs] per minute),
                  ventricular tachycardia or 3rd degree heart block will be excluded from the study
                  unless cleared by a cardiologist

               -  Patients with pulmonary function test abnormalities as evidenced by a (forced
                  expiratory volume 1 [FEV1]/forced vital capacity [FVC ] < 70% of predicted for
                  normality will be excluded

          -  Bone marrow involvement based on PET/CT scan at screening

          -  Active or recent herpes simplex virus (HSV) infection or cytomegalovirus (CMV) based
             on symptoms with positive swab culture and/or positive IgM (immunoglobulin M)
             screening

          -  Liver metastases with no other metastatic sites