CUDC-907 Treatment in People With Metastatic and Locally Advanced Thyroid Cancer

* INCLUSION CRITERIA: * Subjects greater than or equal to 18 years of age. * Thyroid cancer histology or cytology that is aggressive (anaplastic/undifferentiated thyroid cancer, poorly differentiated thyroid cancer, Hurthle cell carcinoma, tall-cell variant of papillary thyroid cancer, sclerosing variant of papillary thyroid cancer). * Measurable disease. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2. * Absolute neutrophil count greater than or equal to 1,000/microL * Platelets greater than or equal to 75,000/microL * Creatinine less than or equal to 1.5 times upper limit of normal (ULN) or creatinine clearance \> 60ml for patients with creatinine levels 1.5 times above institutional ULN (calculated based on age, weight and sex * Total bilirubin less than or equal to 1.5 times ULN; aspartate aminotransferase (AST)/alanine aminotransferase (ALT) less than or equal to 2.5 times ULN. For subjects with documented liver metastases, the AST/ALT may be less than or equal to 5 times ULN. * Recovery to Grade 1 or baseline of any toxicity due to prior anticancer therapies (excluding alopecia). * Platelet transfusions to help patients meet eligibility criteria are not allowed within 3 days prior to screening complete blood count (CBC) or Cycle 1, Day 1 treatment. * Women of child bearing potential must have a negative serum pregnancy test. * The effects of CUDC-907 on the developing human fetus are unknown. For this reason, 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 following the last study treatment. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. -Able to provide written informed consent and to follow protocol requirements. EXCLUSION CRITERIA: * Systemic anticancer therapy within 4 weeks of study entry, except for subjects with anaplastic/undifferentiated thyroid cancer who may be enrolled immediately after discontinuation of previous therapy. * Other investigational agents within 4 weeks prior to study treatment except for subjects with anaplastic/undifferentiated thyroid cancer who may be enrolled immediately of discontinuation of previous therapy. * Pregnant women are excluded from this study because the potential risk of teratogenic or abortifacient effects of CUDC-907 is unknown. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with CUDC-907, breastfeeding should be discontinued if the mother is treated with CUDC-907. These potential risks may also apply to other agents used in this study. * Diabetes mellitus that is not controlled with medication. * Serious infection requiring intravenous antibiotic therapy within 14 days prior to study treatment. * Evidence of central nervous system metastasis. * Uncontrolled or severe cardiovascular disease, including myocardial infarction, unstable angina, or atrial fibrillation (AFib) within 6 months prior to study treatment, New York Heart Association (NYHA) Class II or greater congestive heart failure, serious arrhythmias requiring medication for treatment, clinically significant pericardial disease, cardiac amyloidosis, or corrected QT interval (QTc) with Fridericia's (QTcF) correction that is unmeasurable or greater than or equal to 480 msec on screening electrocardiogram (ECG). (Note: for QTcF greater than or equal to 480 sec on the screening ECG, the ECG may be repeated twice at least 24 hours apart; the mean QTcF from the three screening ECGs must be \< 480 msec in order to meet eligibility for trial participation). * Gastrointestinal disease or disorder that could interfere with the swallowing, oral absorption, or tolerance of CUDC-907. This includes uncontrolled diarrhea (\> 1 watery stool/day), major abdominal surgery, significant bowel obstruction and/or gastrointestinal diseases that could alter the assessment of pharmacokinetics or safety, including but not limited to: irritable bowel syndrome, ulcerative colitis, Crohn's disease and hemorrhagic coloproctitis. * Unstable or clinically significant concurrent medical condition that would, in the opinion of the investigator, jeopardize the safety of a subject and/or compliance with the protocol. * Second primary malignancy within 2 years of study entry other than adequately treated non-melanoma skin or superficial bladder cancer, curatively treated carcinoma in situ of the cervix or other curatively treated solid tumor deemed by the investigator to be at low risk for recurrence.