Testing Lenvatinib in Patients With Adenoid Cystic Carcinoma

Inclusion Criteria: * Patients must have pathologically or cytologically confirmed adenoid cystic carcinoma. Cancers arising from non-salivary gland primary sites are allowed. * Patients must have recurrent and/or metastatic disease not amenable to potentially curative surgery or radiotherapy. * At least 2 weeks must have elapsed since the end of prior systemic treatment (4 weeks for bevacizumab- containing regimens) or radiotherapy with resolution of all treatment-related toxicity to NCI CTCAE Version 4.0 grade ≤1 (or tolerable grade 2) or back to baseline (except for alopecia, lymphopenia, or hypothyroidism). Any number of prior therapies for recurrent/metastatic ACC are allowed. * Patients must have RECIST v1.1 measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral CT scan. * Patients must have documentation of a new or progressive lesion on a radiologic imaging study performed within 6 months prior to study enrollment (progression of disease over any interval is allowed) and/or new/worsening disease related symptoms within 6 months prior to study enrollment. Note: This assessment will be performed by the treating investigator. Evidence of progression by RECIST criteria is not required. * Patients must have archival tissue from the primary tumor or metastases available for correlative studies. Either a paraffin block or twenty unstained slides are acceptable. (If less than twenty unstained slides are available, the patient may be able to participate at the discretion of the investigator.) * Age ≥ 18 years * ECOG performance status \<2 (Karnofsky \>60%) * Adequate bone marrow, liver and renal function (as suggested for lenvatinib studies). * Total bilirubin ≤ 1.5 x the upper limits of normal (ULN) * Alanine aminotransferase (ALT) and aspartate amino-transferase (AST) ≤ 2.5 x ULN (≤ 5 x ULN for subjects with liver involvement of their cancer) * Alkaline phosphatase limit ≤ 2.5 x ULN (≤ 5 x ULN for subjects with liver involvement of their cancer) * Serum creatinine ≤ 1.5 x the ULN or calculated creatinine clearance ≥ 60 ml/min * Platelet count ≥ 100,000 /mm3, hemoglobin (Hb) ≥ 9 g/dL, absolute neutrophil count (ANC) ≥ 1500/mm3. Blood transfusion to meet the inclusion criteria will not be allowed. * Subjects must be able to understand and be willing to sign the written informed consent form. A signed informed consent form must be appropriately obtained prior to the conduct of any trial-specific procedure. * Women of childbearing potential must have a negative serum pregnancy test performed within 2 weeks prior to the start of study drug. Post-menopausal women (defined as no menses for at least 1 year) and surgically sterilized women are not required to undergo a pregnancy test. * Subjects (men and women) of childbearing potential must agree to use adequate contraception beginning at the signing of the consent form until at least 3 months after the last dose of study drug. The definition of adequate contraception will be based on the judgment of the principal investigator or a designated associate. * Subject must be able to swallow and retain oral medication. Exclusion Criteria: * Concurrent anti-cancer therapy (chemotherapy, definitive radiation therapy, surgery, immunotherapy, biologic therapy, or tumor embolization) other than study treatment. Concurrent therapy with bisphosphonates or denosumab for bone metastases is allowed. Palliative radiation to non-target lesions is also allowed. * Prior use of lenvatinib. * Uncontrolled hypertension (systolic pressure \>140 mm Hg or diastolic pressure \> 90 mm Hg \[NCI-CTCAE v4.0\] on repeated measurement) despite optimal medical management. * Concurrent use of another investigational drug or device (i.e., outside of study treatment) during, or within 4 weeks of trial entry (signing of the informed consent form). * Clinically significant proteinuria: * Subjects having \>1+ proteinuria on urinalysis will undergo 24-hour urine collection for quantitative assessment of proteinuria. Subjects with urine protein ≥1 g/24-hour will be ineligible. * Active or clinically significant cardiac disease including: * Congestive heart failure - New York Heart Association (NYHA) \> Class II. * Active coronary artery disease that is not medically treated. * Cardiac arrhythmias requiring anti-arrhythmic therapy other than beta blockers or digoxin. * Unstable angina (anginal symptoms at rest), new-onset angina within 3 months before registration, or myocardial infarction within 6 months before registration. * Subjects with thrombotic, embolic, venous, or arterial events, such as cerebrovascular accident (including transient ischemic attacks) deep vein thrombosis or pulmonary embolism within 6 months of study treatment start. * Symptomatic metastatic brain or leptomeningeal tumors (asymptomatic or treated metastatic brain or leptomeningeal tumors are allowed). * Major surgical procedure, open biopsy, or significant traumatic injury within 28 days before start of study medication. * Evidence or history of bleeding diathesis or coagulopathy. * Any hemorrhage or bleeding event ≥ NCI CTCAE Grade 3 within 4 weeks prior to start of study medication. * Therapeutic anticoagulation with Vitamin-K antagonists (e.g., warfarin) is not allowed if the medication dose and/or INR/PTT are not considered stable by the treating physician. If the dose and/or INR/PTT are stable, therapeutic anticoagulation with Vitamin-K antagonists is allowed with close monitoring. Anticoagulation with heparin or heparinoids is allowed. * Known history of human immunodeficiency virus (HIV) infection or current chronic or active hepatitis B or C infection requiring treatment with antiviral therapy. * Active infection that would impair the ability of the patient to receive study treatment * Presence of a non-healing wound or non-healing ulcer that is not tumor related. * Renal failure requiring hemo-or peritoneal dialysis. * Patients with seizure disorder requiring medication. * Interstitial lung disease with ongoing signs and symptoms at the time of informed consent. * History of organ allograft (including corneal transplant). * Any malabsorption condition. * Women who are pregnant or breast-feeding. * Any condition which, in the investigator's opinion, makes the subject unsuitable for trial participation.