Inclusion Criteria

1. Subjects must have histologically or cytologically confirmed diagnoses of one of the following DTC subtypes: a. Papillary thyroid cancer (PTC) i. Follicular variant ii. Variants (including but not limited to: tall cell, columnar cell, cribriform-morular, solid, oxyphil, Warthin's-like, trabecular, tumor with nodular fasciitis-like stroma, Hurthle cell variant of papillary carcinoma, poorly differentiated); b.Follicular thyroid cancer (FTC) i. Hurthle cell ii. Clear cell iii.Insular

2. Subjects must be radioiodine-refractory or resistant within 12 months of radioiodine therapy and have one of the following: 1. One or more lesions that do not demonstrate iodine uptake on any radioiodine scan 2. One or more lesions that have substantially increased in size within 12 months of radioiodine therapy, despite demonstration of radioiodine activity at the time of that treatment by pre- or posttreatment scanning 3. Cumulative activity of radioiodine of greater than 600 mCi or 22 gigabecquerels (GBq), with the last dose administered at least 6 months prior to study entry

3. Subjects must have received and failed treatment with sorafenib for RR-DTC. This criterion will not apply outside the United States if sorafenib is not commercially available in the country where the subject resides.

4. ECOG performance status of 0 to 2

5. Blood pressure (BP) less than or equal to 150/90 mmHg at screening with or without antihypertensive medications and no change in antihypertensive medications within 1 week prior to Cycle 1 Day 1

6. Creatinine clearance greater than or equal to 30 mL/min according to the Cockcroft and Gault formula

7. Adequate bone marrow function - Absolute neutrophil count (ANC) greater than or equal to 1.5 x 10^9/L - Hemoglobin greater than or equal to 9.0 g/dL (can be corrected by growth factor or transfusion) - Platelet count greater than or equal to 100 x 10^9/L

8. Adequate liver function - Bilirubin less than or equal to 1.5 x upper limit of normal (ULN) except for unconjugated hyperbilirubinemia or Gilbert's syndrome - Alkaline phosphatase (ALP), alanine aminotransferase (ALT), and aspartate aminotransferase (AST) less than or equal to 3 x ULN (less than or equal to 5 x ULN if subject has liver metastases). If ALP is greater than 3 x ULN (in the absence of liver metastases) or greater than 5 x ULN (in the presence of liver metastases) AND subjects are also known to have bone metastases, the liver-specific ALP must be separated from the total and used to assess the liver function instead of the total ALP

9. Provide written informed consent

10. Males and females age greater than or equal to 18 years at the time of informed consent

11. All females must have a negative serum or urine pregnancy test. Females of childbearing potential and male subjects who are partners of women of childbearing potential must use or their partners must use a highly effective method of contraception

Exclusion Criteria

1. Subjects having greater than 1+ proteinuria on urine dipstick at screening testing will undergo 24 h urine collection for quantitative assessment of proteinuria. Subjects with urine protein greater than or equal to 1 g/24 h will be ineligible.

2. History of congestive heart failure with New York Heart Association (NYHA) Classification greater than II, unstable angina, myocardial infarction, serious cardiac arrhythmia, or stroke within the past 6 months

3. Electrocardiogram (ECG) with QT interval (QTc) interval greater than or equal to 480 msec

4. Existing anti-cancer therapy-related toxicities of grade greater than or equal to 2, except alopecia and infertility

5. History of intolerance to or progression on prior treatment with lenvatinib that led to the discontinuation of lenvatinib

6. Any history of or concomitant medical condition that, in the opinion of the investigator, would compromise subject's ability to safely complete the protocol

7. Females who are pregnant (positive B-hCG test) or breastfeeding

8. Eligible for any other lenvatinib study that is open for recruitment.