Prospective Randomized Trial Comparing Gastrectomy, Metastasectomy Plus Systemic Therapy Versus Systemic Therapy Alone: GYMSSA Trial

-INCLUSION CRITERIA: 1. Histologically or cytologically confirmed gastric adenocarcinoma. 2. Metastatic disease must be measurable by computed tomography (CT) and/or magnetic resonance imaging (MRI) Or There must be a history of positive peritoneal washings or carcinomatosis 3. All disease should be deemed resectable to negative margins (NED) based on imaging studies. Note: Patients with both pulmonary and peritoneal metastases will be enrolled at the discretion of the Principal Investigator. * Esophageal invasion \< 4cm that does not require thoracotomy (Seiwert II and III lesions). * Hepatic metastases (unilateral or bilateral less than or equal to 5 lesions, less than or equal to 15 cm total diameter). * Primary peritoneal metastases (small disease load less than or equal to P2 disease) without massive ascites or intestinal obstruction. * Para-aortic lymph node metastases (stations 16 a1 and/or b2). * Lung metastases (less than or equal to 3 unilateral/bilateral, 9 cm total diameter). * Patients who present with both hepatic and peritoneal metastases must have no evidence of extensive para-aortic/retro-pancreatic lymph node metastases. 4. Greater than or equal to 18 years of age. 5. Must be able to understand and sign the Informed Consent Document. 6. Clinical performance status of Eastern Cooperative Oncology Group (ECOG) less than or equal to 2. 7. Life expectancy of greater than three months. 8. Patients of both genders must be willing to practice birth control during and for four months after receiving chemotherapy. 9. Hematology: * Absolute neutrophil count greater than 1300/mm\^3 without the support of Filgrastim. * Platelet count greater than 75,000/mm\^3. * Hemoglobin greater than 8.0 g/dl. 10. Chemistry: * Serum alanine aminotransferase (ALT)/aspartate aminotransferase (AST) less than or equal to 5 times the upper limit of normal. Except in the presence of obstructive liver metastases where ALT/AST may be up to 10 times the upper limit of normal. * Serum creatinine less than or equal to 1.5 mg/dl unless the measured creatinine clearance is greater than 60 mL/min/1.73 m\^2. * Total bilirubin less than or equal to 2 mg/dl, except in the presence of obstructive metastases. * Prothrombin time (PT) within 2 seconds of the upper limit of normal (International Normalized Ratio (INR) less than or equal to 1.8). 11. No history of prior/other malignancies within the 2 years prior to enrollment with the exception of basal cell carcinoma. EXCLUSION CRITERIA: 1. Prior treatment with 5-FU, leucovorin, oxaliplatin, and irinotecan (FOLFOXIRI) (treatment with any of the components as separate regimens is allowable). 2. Inability to tolerate any of the chemotherapeutic agents. 3. Grade 2 or greater neuropathy. 4. Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the chemotherapy on the fetus or infant. 5. Active systemic infections, coagulation disorders or other major medical illnesses of the cardiovascular, respiratory or immune system; myocardial infarction; cardiac arrhythmias; obstructive or restrictive pulmonary disease. 6. Brain metastases or a history of brain metastases. 7. Childs B or C cirrhosis or with evidence of severe portal hypertension by history, endoscopy, or radiologic studies. 8. Weight less than 40 kg. 9. Significant ascites, greater than 1000cc in the absence of peritoneal disease. 10. History of congestive heart failure and/or an left ventricular ejection fraction (LVEF) less than 40%. Note: Patients at increased risk for coronary artery disease or cardiac dysfunction (e.g., greater than 65yo, diabetes, history of hypertension, elevated low-density lipoprotein (LDL), first degree relative with coronary artery disease) will undergo full cardiac evaluation and will not be eligible if they demonstrate significant irreversible ischemia on stress thallium or an ejection fraction \< 40%. 11. Significant chronic obstructive pulmonary disease (COPD) or other chronic pulmonary restrictive disease with pulmonary function tests (PFT's) indicating an forced expiratory volume 1 (FEV1) less than 50% or a carbon monoxide diffusing capacity (DLCO) less than 40% predicted for age. Note: Patients who have shortness of breath with minimal exertion or who are at risk for pulmonary disease (e.g., chronic smokers) will undergo pulmonary function testing and will not be eligible if their FEV1 is less than 50% of expected.