The goal of this clinical research study is to find the highest tolerable dose of paclitaxel that can be given as hyperthermic intraperitoneal chemotherapy (HIPEC) to patients with gastric or gastroesophageal cancer. HIPEC is a system in which heated chemotherapy is delivered directly inside the abdomen during surgery. In this study, paclitaxel is being combined with mitomycin and cisplatin to see if this study drug combination can help to control the disease.

This is an investigational study. Mitomycin, cisplatin, and paclitaxel are FDA-approved and commercially available for the treatment of gastric and gastroesophageal cancer. It is investigational to give these drugs by HIPEC.

The study doctor can describe how the study drugs and HIPEC are designed to work.

Up to 48 participants will be enrolled in this study. All will take part at MD Anderson.

Study Groups:

If participant is found to be eligible to take part in this study, participant will be assigned to a dose level of paclitaxel based on when participant joins this study. Up to 8 dose levels of paclitaxel will be tested. Up to 6 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of paclitaxel is found.

All participants will receive the same dose level of mitomycin and cisplatin.

HIPEC Treatment and Study Visits:

On the day of HIPEC treatment, participant will receive heated mitomycin, cisplatin, and paclitaxel as a liquid that is injected through 3 to 4 small incisions in participant's abdomen over about 1 hour. Heated mitomycin, cisplatin, and paclitaxel will be delivered through plastic tubing that is connected to a pump into the abdominal cavity. The pump pushes the heated drugs into the abdominal cavity and then pulls it out and recirculates the drugs. The skin of the abdominal cavity is temporarily closed during this procedure. A pump will be used to pump the heated drugs in and out of the abdominal cavity over 60 minutes while the surgeon gently presses on the abdominal wall so the drugs can reach all areas in the abdominal cavity. After 60 minutes, the drugs are removed and the abdominal cavity will be washed. The remaining fluid will be removed before the surgeon closes the abdominal cavity with stitches.

Additionally, participant will be given standard drugs to help decrease the risk of side effects. Participant may ask the study staff for information about how the drugs are given and their risks.

Participant will remain in the hospital for 3-7 days after treatment.

While participant is in the hospital after surgery, blood (about 2-3 tablespoons) may be drawn for routine tests at any time that the study doctor thinks it is needed.

One (1) time between 2-6 weeks after HIPEC Treatment:

• Participant will have a physical exam.
• Blood (about 2-3 tablespoons) will be drawn for routine tests.

Follow-Up Visits:

Participant will have routine clinic visits or participant will be called every 6 months after the HIPEC procedure. Participant will be asked about any other cancer treatments participant may be receiving. If participant is called, these calls should last about 5-10 minutes. If participant stopped the study early, the study doctor may ask participant to return to the clinic for extra CT scans, PET scan, or MRIs during the follow-up period.

Participant will have a CT scan, PET scan, or MRI of participant's chest, abdomen, and pelvis every 6 months for 5 years after surgery to check the status of the disease.

Length of Study Participation:

Participant will be on study for about 5 years after participant's last procedure (HIPEC or surgery). Participant will be taken off study early if the disease gets worse, if intolerable side effects occur, or if participant is unable to follow study directions.

Participation on the study will be over after the follow-up visits.

Inclusion Criteria:

1. Age 18 years and above. There will be no upper age restriction.
2. Eastern Cooperative Oncology Group (ECOG) performance status </= 2.
3. Cytologic or histologic proof of adenocarcinoma of the stomach or gastroesophageal junction.
4. Adequate renal, and bone marrow function: a. Leukocytes >= 3,000/uL b. Absolute neutrophil count >= 1,500/uL c. Platelets >= 60,000/Ul d. Serum creatinine <= 1.5 mg/dL
5. Distant Metastatic Disease of peritoneum: a. Positive peritoneal cytology b. Carcinomatosis on diagnostic laparoscopy or laparotomy.
6. Completion of preoperative systemic chemotherapy.

Exclusion Criteria:

1. Infections such as pneumonia or wound infections that would preclude protocol therapy.
2. Women with a positive urine or serum pregnancy test are excluded from this study; women of childbearing potential (defined as those who have not undergone a hysterectomy or who have not been postmenopausal for at least 24 consecutive months) must agree to refrain from breast feeding and practice adequate contraception as specified in the informed consent. Adequate contraception consists of oral contraceptive, implantable contraceptives, injectable contraceptives, a double barrier method, or abstinence.
3. Subjects with unstable angina or New York Heart Association Grade II or greater congestive heart failure.
4. Subjects deemed unable to comply with study and/or follow-up procedures.
5. Subjects with a known hypersensitivity to protocol systemic chemotherapy that was life-threatening, required hospitalization or prolongation of existing hospitalization, or resulted in persistent or significant disability or incapacity.