The goal of this clinical research study is to find the highest tolerable dose of the combination of dasatinib, bevacizumab, and paclitaxel with or without Methylnaltrexone that can be given to patients with advanced cancer. The safety of this drug combination will also be studied.

The Study Drugs:

Dasatinib is designed to decrease the activity of one or more proteins that are responsible for the uncontrolled growth of tumor cells. This may cause the tumor cells to die.

Bevacizumab is designed to block the growth of blood vessels that supply nutrients necessary for tumor growth. This may prevent or slow down the growth of cancer cells.

Paclitaxel is designed to block cancer cells from dividing, which may cause them to die.

Methylnaltrexone is designed to block the constipating effect of opioid analgesics on the gastrointestinal tract. It is also capable of blocking the growth of blood vessels that supply nutrients necessary for tumor growth. This may prevent or slow down the growth of cancer cells.

Study Groups:

Dose escalation:

If you are found to be eligible to take part in this study, you will be assigned to a dose level of dasatinib, bevacizumab, and paclitaxel based on when you join this study. Up to 8 dose levels of dasatinib, bevacizumab, and paclitaxel will be tested. Three (3) to 9 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 the combination of dasatinib, bevacizumab, and paclitaxel is found.

The highest tolerable dose of the combination of dasatinib, bevacizumab, and paclitaxel that was found will be given in combination with methylnaltrexone to up to 6 participants. If intolerable side effects occur, the next group of up to 6 participants will receive a lower dose of methylnaltrexone but the same dose of the combination of dasatinib, bevacizumab, and paclitaxel.

Dose expansion:

Once the highest safe dose of the combinations of dasatinib, bevacizumab, and paclitaxel with and without methylnaltrexone are found, 14 additional participants will be enrolled and receive the study drugs at each dose level combination. Participants with a tumor type (14 for each tumor type) that have responded to the study drug combination will receive the study drugs at that dose level, as well.

Study Drug Administration:

Each study "cycle" is 28 days.

Everyday, you will take dasatinib by mouth 1 time a day. You should take it at about the same time each day with food and a cup of water (about 8 ounces).

On Days 1 and 15 of each cycle, you will receive bevacizumab by vein over 90 minutes. If the first dose is well tolerated, you will receive the next dose over 60 minutes. If the second dose is well tolerated, you will receive the next doses over 30 minutes.

On Days 1, 8, and 15 of each cycle, you will receive paclitaxel by vein over 60 minutes. On Days 1 and 15, your paclitaxel dose will be given after your bevacizumab dose.

About 30 minutes before each scheduled dose of paclitaxel, you will also receive medications (such as dexamethasone) to lower the likelihood of experiencing allergic reactions.

If you receive methylnaltrexone, you will receive it by injection under the skin either to the upper arm, upper thigh, or stomach at the same time 2 times every day. The study doctor or nurse will teach you how to give the injections yourself.

Study Visits:

At every study visit, you will be asked about any current health conditions you have, drugs you may be taking, and if you have experienced any side effects.

Around Days 8 and 28 of Cycle 1:

• You will have a physical exam.
• Blood (about 2 teaspoons) will be drawn for routine tests.

Around Day 15 of Cycle 1, blood (about 2 teaspoons) will be drawn for routine tests.

Around Day 28 of Cycles 2 and beyond:

• You will have a physical exam.
• Blood (about 2 teaspoons) will be drawn for routine tests.

Every 4 weeks, you will have a blood (about 1 teaspoon) drawn or urine collected for pregnancy test if you are able to become pregnant.

Every 8 weeks, you will have an x-ray, CT scan, MRI scan, and/or PET/CT scan to check the status of the disease. If the study doctor thinks it is needed, they will be performed more often.

Length of Study:

You may stay on study for as long as the disease does not get worse, you have not experienced intolerable side effects, and if the study doctor thinks it is in your best interest.

End-of-Study Visit:

About 28 days after the last dose of study drugs, you will have an end-of-study visit. At this visit, the following tests or procedures may be performed:

• You will have a physical exam, including measurement of your vital signs.
• Your performance status will be recorded.
• Blood (about 2 teaspoons) will be drawn for routine tests.
• If the doctor thinks it is needed, you will have an x-ray, CT scan, MRI scan, and/or PET/CT scan to check the status of the disease.

This is an investigational study. Dasatinib is FDA approved and commercially available for the treatment of chronic myeloid leukemia. Bevacizumab is FDA approved and commercially available for the treatment of colorectal, breast, lung, and brain cancer. Paclitaxel is FDA approved and commercially available for the treatment of breast, lung, and ovarian cancer and Kaposi's sarcoma. Methylnaltrexone is FDA approved and commercially available for the treatment of constipation in patients with advanced illness who are receiving palliative care. The combination of these drugs when given to patients with advanced cancer is investigational.

Up to 218 patients will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients with advanced or metastatic cancer that is refractory to standard therapy, relapsed after standard therapy, or who have no standard therapy available that improves survival by at least three months.
2. Patients must be >/= 3 weeks beyond treatment with a cytotoxic chemotherapy regimen, or therapeutic radiation, or major surgery. Patients may have received palliative localized radiation immediately before or during treatment providing radiation is not delivered to the only site of disease being treated under this protocol. After targeted/biologic therapy a patient has to be off treatment for 5 half-lives or 3 weeks whatever is shorter.
3. ECOG performance status </= 2.
4. Patients must have normal organ and marrow function defined as: absolute neutrophil count >/= 1,000/mL; platelets >/=90,000/mL; creatinine </= 2 X ULN; total bilirubin </= 2.0; ALT(SGPT) </= 5 X ULN; Exception for patients with liver metastasis: total bilirubin </= 3 x ULN; ALT(SGPT) </= 8 X ULN.
5. Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 30 days after the last dose.
6. Patients must be able to understand and be willing to sign a written informed consent document.

Exclusion Criteria:

1. Uncontrolled intercurrent illness including, but not limited to, uncontrolled infection, uncontrolled asthma, need for hemodialysis, need for ventilatory support.
2. Patients with hemoptysis within 28 days prior to entering the study.
3. Patients with clinically significant unexplained bleeding within 28 days prior to the first dose of study medication.
4. Uncontrolled systemic vascular hypertension (systolic blood pressure > 140mmHg, diastolic blood pressure > 90mmHg on medication).
5. Patients with clinically significant cardiovascular disease: history of CVA within 6 months; myocardial infarction or unstable angina within 6 months.
6. Major surgery within 28 days prior to Day 1 of dosing Bevacizumab.
7. Pregnant or lactating women.
8. History of hypersensitivity to dasatinib or any component of the formulation.
9. History of hypersensitivity to bevacizumab, murine products, or any component of the formulation.
10. History of hypersensitivity to paclitaxel or any component of the formulation.
11. Patients with pleural effusion which is considered clinically significant by the attending physician.
12. Patients unwilling or unable to sign informed consent document.
13. Social situations that would limit compliance with study requirements.
14. Patients receiving opioids within 2 weeks before signing the consent and patients, who cannot be off opioids until initiating the study medication (for methylnaltrexone arm only).