The goal of this clinical research study is to find the highest tolerable dose of the combination of regorafenib and cetuximab that can be given to patients with advanced solid tumors. The safety and effectiveness of this drug combination will also be studied.

Study Groups:

If you are found to be eligible to take part in this study, you will be assigned to a dose level of the study drugs based on when you join the study.

Up to 5 dose level combinations of regorafenib and cetuximab will be tested. Up to 3 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level combination. Each new group will receive a higher dose of either regorafenib or cetuximab than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose combination of regorafenib and cetuximab are found.

Once the highest tolerable dose combination is found, up to 10 more participants may be enrolled. This group (called the Dose Expansion group) is enrolled to help researchers study the safety and effectiveness of this dose level combination.

Study Drug Administration:

Each study cycle is 28 days.

You will take regorafenib by mouth on Days 1-5, 8-12, and 15-19 of each cycle. You will swallow the whole capsules with 1 cup of water (about 8 ounces) 1 time a day. Do not break, chew, or open your capsules. The doctor will discuss this with you.

You will receive cetuximab by vein over about 1-2 hours on Days 1, 8, 15, and 22 of each cycle.

Study Visits:

At each cycle, you will be asked to complete a questionnaire so the study team will better understand any symptoms you may be having that are related to use of the study drug.

On Day 1 of Cycle 1, blood (about 4 teaspoons) and urine will be collected for routine tests and pharmacodynamic (PD) tests. PD testing measures how the level of study drug in your body may affect the disease.

Between Days 8 and 14, Days 15 and 21, and Days 22 and 28 of Cycle 1:

°Blood (about 4 teaspoons) will be drawn for routine tests.

On Day 1 of Cycles 2 and beyond:

• You will have a physical exam.
• Urine will be collected for routine tests.

Once a week, blood (about 4 teaspoons) will be drawn for routine tests.

Every other cycle (every 8 weeks):

°You will have an x-ray and a CT scan to check the status of the disease.

Length of Study:

You may continue receiving regorafenib and cetuximab for as long as the doctor thinks it is in your best interest. You will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if you are unable to follow study directions.

You will be off study after the end-of-study visit.

End of Study Visit:

About 30 days after your last dose of study drugs:

• You will have a physical exam.
• Blood (about 4 teaspoons) and urine will be collected for routine and PD tests tests.
• You will have an x-ray and a CT scan to check the status of the disease.

Follow-up:

You will be asked about any health problems you may have and if you have had any side effects. If your study doctor thinks it is needed, you may have follow-up for a longer period of time.

This is an investigational study. Regorafenib is FDA approved for the treatment of metastatic colorectal cancer. Cetuximab is FDA approved for the treatment of K-Ras wild type colorectal cancer as well as head and neck cancer. The combination of regorafenib and cetuximab is currently being used in advanced solid tumor patients for research purposes only.

Up to 83 patients will be enrolled in this study. All will take part at MD Anderson.

Inclusion Criteria:

1. Patient must be >= 12 years of age and > 40kgs.
2. Patients with advanced or metastatic cancer that is refractory to standard therapy, relapsed after standard therapy, or have no standard therapy that induces a CR rate of at least 10% or improves survival by at least three months.
3. Patients must have evaluable or measurable disease by RECIST criteria for solid tumors.
4. Ability to understand and the willingness to sign a written informed consent document.
5. Patients must be >/= 4 weeks beyond treatment with any chemotherapy or radiotherapy, and must have recovered to </= grade 2 toxicity for any treatment-limiting toxicity of prior therapy. (Exception: patients may have received palliative low dose radiotherapy to the limbs 1-4 weeks before this therapy provided pelvis, ribs, sternum, scapulae, vertebrae or skull were not included in the radiotherapy field). Also, patients who have received non-chemotherapeutic biologic agents will need to wait at least five half-lives or four weeks, whichever is shorter, from the last day of treatment. Exception: No washout of cetuximab or regorafenib is required for patients who have received prior cetuximab or regorafenib and have recovered from any treatment-related toxicities to Grade </= 1.
6. ECOG performance status </= 2 (Karnofsky >/= 60%).
7. Patients must have: leukocytes >/= 3,000/mL; absolute neutrophil count >/= 1,000/mL; platelets >/=100,000/mL; creatinine </= 2 X ULN; total bilirubin </= 2.0; ALT(SGPT) </= 3 X ULN; Exception for patients with liver metastasis: total bilirubin </= 3 x ULN; AST (SGOT) and ALT(SGPT) </= 5 X ULN. Patients should not have received any platelet transfusions in the last 4 weeks before screening date.
8. Patients should not become pregnant or breastfeed while on this study. Sexually active patients must agree to use contraception prior to study entry, for the duration of study participation, and for 30 days after the last dose.

Exclusion Criteria:

1. KRAS mutated colorectal cancer.
2. Major surgery within 28 days prior to the first dose of study medication.
3. Pregnant or lactating women.
4. Patients with hemoptysis within 28 days prior to entering the study.
5. Patients with clinically significant unexplained bleeding within 28 days prior to entering the study.
6. Uncontrolled systemic vascular hypertension (Systolic blood pressure > 140 mmHg, diastolic blood pressure > 90 mmHg on medication).
7. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring parenteral antibiotics on Day 1.
8. Patients with clinically significant cardiovascular disease: 1). History of CVA within 6 months; 2). Myocardial infarction or unstable angina within 6 months; 3). Unstable angina pectoris; 4). New York Heart Association Class III or greater congestive heart failure.
9. Patients with untreated or progressing brain metastases.
10. Patients who had radiation to greater than 25% marrow in the past 5 years.