There are 2 phases in this study: Phase 1 (dose escalation) and Phase 2 (dose expansion).

The goal of Phase 1 of this clinical research study is to find the highest tolerable dose of lenvatinib and Xeloda (capecitabine) that can be given to patients with advanced cancer. The goal of Phase 2 of this study is to learn if the dose of lenvatinib and capecitabine found in Phase 1 can help to control advanced cancer.

The safety of this drug combination will be studied in both phases of the study.

Study Groups:

If participant is are found to be eligible to take part in this study, they will be assigned to a study group based on when they join this study. Up to 4 groups of up to 18 participants will be enrolled in Phase 1 of the study, and up to 28 participants will be enrolled in Phase 2.

If participant is enrolled in Phase 1, the dose of lenvatinib they receive will depend on when they join this study. The first group of participants will receive the lowest dose level of lenvatinib. Each new group will receive a higher dose of lenvatinib than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of lenvatinib is found.

If participant is enrolled in Phase 2, they will receive lenvatinib at the highest dose that was tolerated in Phase 1.

All participants will receive the same dose of capecitabine.

Study Drug Administration:

Each study cycle is 21 days.

Participant will take lenvatinib by mouth every day and capecitabine by mouth 2 times every day.

Length of Study Participation:

Participant may continue taking the study drugs for as long as the doctor thinks it is in their best interest. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if they are unable to follow study directions.

Patient's participation on the study will be over after the end-of-study visit (described below).

Study Visits:

During Week 2 of Cycle 1:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) will be drawn for routine tests.
• If participant is enrolled in Phase 2, they will have a tumor biopsy for biomarker testing.
• If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing.

During Week 3 of Cycle 1, blood (about 3 teaspoons) will be drawn for routine tests.

During Week 1 of Cycles 2 and beyond:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) will be drawn for routine tests.
• If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing.
• If participant can become pregnant, blood (about 1 teaspoon) will be collected for a pregnancy test.

During Weeks 2 and 3 of Cycles 2 and beyond, participant will be called by a member of the study staff and asked about any side effects they may have. This call should last about 5-10 minutes.

During Week 3 of Cycle 2 and then every even-numbered cycle after that (Cycles 4, 6, 8, and so on), participant will have a CT scan or MRI.

End-of-Study Visit:

After participant's last dose of study drug, they will have an end-of-study visit. At this visit, the following tests and procedures will be performed:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) will be drawn for routine tests.
• Participant will have a CT scan or MRI to check the status of the disease.
• If participant can become pregnant, blood (about 1 teaspoon) or urine will be collected for a pregnancy test.
• If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing.

This is an investigational study. Lenvatinib is FDA approved and commercially available for the treatment of certain types of thyroid cancer. Capecitabine is FDA approved and commercially available for the treatment of certain types of breast and colorectal cancers. It is considered investigational to use lenvatinib and capecitabine to treat advanced cancer.

The study doctor can explain how the study drugs are designed to work.

Up to 46 participants will be enrolled in this study. All will take part at MD Anderson.

Inclusion Criteria:

1. Patients with a histologically and/or cytologically confirmed solid tumor who are resistant / refractory to approved therapies or for whom no curative therapies are available.
2. All previous treatment (including surgery, radiotherapy and systemic anti-neoplastic therapy) must have been completed at least three weeks prior to study entry and any acute toxicities must have resolved.
3. Aged >/= 18 years.
4. Eastern Cooperative Oncology Group (ECOG) performance status score of </= 2.
5. Written informed consent prior to any study specific screening procedures with the understanding that the patient may withdraw consent at any time without prejudice.
6. Willing and able to comply with the protocol guidelines for the duration of the study.

Exclusion Criteria:

1. Unstable metastases to the central nervous system (CNS).
2. Any of the following laboratory parameters: a) hemoglobin < 9 g/dL (5.6 mmol/L); b) neutrophils <1.5 x 109/L; c) platelets <100 x 109/L; d) serum bilirubin >25 µmol/L (1.5 mg/dL); e) liver function tests with values >3 x upper limit of normal (ULN) f) serum creatinine >1.5 x ULN or creatinine clearance < 60 mL/minute
3. Positive history of HIV, active hepatitis B or active hepatitis C or severe/uncontrolled intercurrent illness or infection
4. Centrally located non-small cell lung cancers and squamous cell lung cancers
5. Clinically significant cardiac impairment or unstable ischemic heart disease including a myocardial infarction within six months of study start
6. Patients with marked Baseline prolongation of QT/QTc interval (QTc interval > 450 msec for males or > 470 msec for females) using the Fridericia method for QTc analysis
7. Bleeding or thrombotic disorders, or using therapeutic dosages of anticoagulants, such as warfarin. Occasional use of NSAIDs and antiplatelet agents such as aspirin, clopidogrel, aggrenox and dipyridamole are not considered exclusionary if taken <7 days per 28 days. However, if the patient requires chronic use (>/=7 days out of 28 days) of full doses of aspirin or NSAIDs then the patient is excluded.
8. Requirement for chronic use of full dose aspirin or non-steroidal anti-inflammatory drugs (NSAIDs)
9. Poorly controlled hypertension (defined as requiring changes in any hypertensive regimen within 1 week of study entry) or patients diagnosed with hypertension based on repeat blood pressure measurements of >160/90 mmHg at Screening
10. Proteinuria > 1+ on urine dipstick testing or 30 mg/dL
11. A history of gastrointestinal malabsorption or having undergone surgery requiring gastrointestinal anastomoses within four weeks of starting therapy or who have not recovered from major surgery within three weeks of starting therapy
12. History of alcoholism, drug addiction, or any psychiatric or psychological condition which, in the opinion of the Investigator, would impair study compliance.
13. Any treatment with investigational drugs within 30 days before the start of the study
14. Previous treatment with E7080
15. Women who are pregnant or breast-feeding; women of childbearing potential with a positive pregnancy test at Screening or no pregnancy test. Women of childbearing potential unless (1) surgically sterile or (2) using adequate measures of contraception (including two forms of contraception, one of which must be a barrier method) in the opinion of the Investigator. Perimenopausal women must be amenorrheic for at least 12 months to be considered of non-childbearing potential.
16. Fertile males with female partners who are not willing to use contraception or whose female partners are not using adequate contraceptive protection
17. Legal incapacity