Patients must have a thyroid-stimulating hormone (TSH) with reflex free T3/free T4 (if TSH is out of normal range) and electrocardiogram (EKG) obtained within 7 days prior to sub-study registration Patients must have adequate thyroid function, as evidenced by either thyroid-stimulating hormone (TSH) or, free thyroxine (T4) serum tests demonstrating values within the normal range, within 28 days prior to registration; Note: TSH, with reflex T4 is allowable if per institutional standard; otherwise, both TSH and free-T4 must be obtained Thyroid stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible Re-registration: TSH WNL; supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH, if free T4 is normal and patient is clinically euthyroid, patient is eligible Thyroid-stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH, if free T4 is normal and patient is clinically euthyroid, patient is eligible Thyroid stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH however if the Free T4 is normal and patient is clinically euthyroid, patient is eligible A serum thyroid-stimulating hormone (TSH) must be obtained within 28 days prior to step 2 registration to obtain a baseline value Thyroid-stimulating hormone (TSH) within normal limits (TSH < ULN allowed in euthyroid patients on thyroid replacement therapy) Adequate organ function as defined by System Laboratory Values; Hematologic (Absolute neutrophil count [ANC] >=1.5x10^9/ liter [L], Lymphocyte count >1,000/cubic millimeter [mm^3], Hemoglobin >=9 grams/deciliter [g/dL], Platelets >=100x10^9/L), Hepatic (Total bilirubin <=1.5x upper limit of normal [ULN] [For subjects with Gilbert's Syndrome, only if direct bilirubin <=35%, <=3.0xULN], alanine aminotransferase [ALT] <=1.5xULN); Renal (Serum Creatinine <=1.5xULN OR Calculated creatinine clearance [CrCl >50 mL/min) and Endocrine (Thyroid stimulating hormone [TSH] within normal limits. If TSH is not within normal limits at baseline, the subject may still be eligible if total T3 or free T3 and free T4 are within the normal limits. Any isolated laboratory abnormality suggestive of a serious autoimmune disease (e.g. hypothyroidism): antinuclear antibody, thyroid-stimulating hormone (TSH), free thyroxine (FT4), rheumatoid factor Thyroid stimulating hormone (TSH) OR free thyroxine (T4) within the normal limits Thyroid stimulating hormone (TSH) =< 1 x ULN. Thyroid-stimulating hormone (TSH) WNL\r\n* Supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH, if free T4 is normal and patient is clinically euthyroid, patient is eligible Adequate thyroid function within 14 days prior to registration defined as serum thyroid-stimulating hormone (TSH) within the normal range Normal serum TSH within 12 months preceding surgery Patients may have had a recent previous hospital admission (within 30 days) or be admitted preoperatively but not for the following conditions\r\n* Unstable angina\r\n* Congestive heart failure\r\n* Severe hypothyroidism thyroid-stimulating hormone (TSH) >10\r\n** Endocrine consult and intervention may allow participation at the discretion of the principal investigator (PI) for a TSH > 10 Triiodothyronine measurement (T3), thyroxine (T4) or thyroid-stimulating hormone (TSH) =< 3.0 x IULN Adequately controlled thyroid function, with no symptoms of thyroid dysfunction and thyroid stimulating hormone (TSH) within normal limits Participants with DTC must be receiving thyroxine suppression therapy and levels of thyroid stimulating hormone (TSH) should not be elevated (TSH should be less than or equal to 5.50 milliunits per liter (mU/L)). When tolerated by the participant, thyroxine dose should be changed to achieve TSH suppression (TSH less than 0.50 mU/L). Thyroid-stimulating hormone (TSH) and free thyroxine (T4) within normal limits (patients may be on thyroid hormone replacement) A serum thyroid-stimulating hormone (TSH), free T4, and AM (morning) cortisol must be obtained within 14 days prior to randomization to obtain a baseline value and be within normal limits for the local laboratory Thyroid stimulating hormone (TSH) within normal limits Adequately controlled thyroid function, with no symptoms of thyroid dysfunction and thyroid-stimulating hormone (TSH) within normal limits Normal thyroid function testing (thyroid stimulating hormone [TSH]) within 14 days prior to registration Thyroid-stimulating hormone (TSH) up to 4 times ULN if thyroxine (T4) is normal Thyroid-stimulating hormone (TSH) within institutional/laboratory normal limits Thyroid stimulating hormone (TSH) within institutional normal range – patients with thyroid disease are eligible if euthyroid on suppressive or replacement therapy Thyroid-stimulating hormone (TSH) within normal limits (WNL); supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH however free T4 and free thyroxine index (FTI) are normal and patient is clinically euthyroid, patient is eligible Thyroid stimulating hormone (TSH) within normal limits Normal (=< upper limit of normal) levels of thyroid stimulating hormone (TSH), free T4, adrenocorticotropic hormone (ACTH), morning AM cortisol, amylase and lipase are required for entry to cohorts 4, 5 and 6 (AM cortisol testing may be taken up to 5 pm per Moffit Cancer Center guidelines) Patients should have normal baseline thyroid-stimulating hormone (TSH); patients with elevated TSH level (between 4.5-10 mU/L in the absence of symptoms of hypothyroidism) must have a normal free thyroxine (T4); TSH that is 10 mU/L or higher is exclusionary\r\n* A history of hypothyroidism and/or hyperthyroidism is allowed, as long as the patient has stable well-controlled thyroid function for a minimum of 2 months Thyroid stimulating hormone (TSH) within institutional limits (ie: normal); if TSH is greater or less than institutional limits patients may participate if their T4 is within normal limits (WNL); patients may be on a stable dose of replacement thyroid medication; dose adjustments are allowed if needed Adequate thyroid function within 28 days prior to registration defined as serum thyroid-stimulating hormone (TSH) in normal range Thyroid-stimulating hormone (TSH) (0.3-5.0) and free thyroxine (0.8-1.87) within Mayo normal range Uncompensated hypothyroidism (defined as thyroid-stimulating hormone [TSH] greater than 2 x upper limit of normal not treated with replacement hormone) Adequately controlled thyroid function, with no symptoms of thyroid dysfunction and thyroid stimulating hormone (TSH) within normal limits Patients must have a normal baseline thyroid stimulating hormone (TSH); a history of hypothyroidism and/or hyperthyroidism is allowed Thyroid stimulating hormone (TSH) within institutional limits (i.e.: normal); if TSH is greater or less than institutional limits patients may participate if their T4 is within normal limits (WNL); patients may be on a stable dose of replacement thyroid medication; dose adjustments are allowed if needed (labs should be performed within 14 days of treatment initiation) Uncontrolled hypothyroidism; patients must have a normal thyroid-stimulating hormone (TSH) per institutional standards at baseline Thyroid stimulating hormone (TSH) within normal limits (WNL); medications for thyroid dysfunction are allowed as long as TSH is normal at registration; in patients with abnormal TSH, if the free thyroxine (free T4) and free thyroxine index (FTI) are normal and patient is clinically euthyroid, patient is eligible Any ongoing toxicity from prior anti-cancer therapy that is > grade 1 and/or that is progressing in severity; hypothyroidism treated with medication is not excluded if thyroid-stimulating hormone (TSH) is within normal limits The patient has normal thyroid function tests (thyroid-stimulating hormone [TSH], free T4) as defined by the testing laboratory, a test abnormality that is asymptomatic and does not warrant medical intervention, or a pre-existing thyroid disorder that is controlled on medical treatment Thyroid-stimulating hormone (TSH) >= LLN and free T4 within normal limits; patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism; the thyroid supplement dose must be a stable dose for at least one month prior to study enrollment Performed within 10 days of treatment initiation: Thyroid stimulating hormone (TSH) within institutional limits (ie: normal); if TSH is greater or less than institutional limits patients may participate if their thyroxine (T4) is within normal limits (WNL); patients may be on a stable dose of replacement thyroid medication; dose adjustments are allowed if needed Uncontrolled hypothyroidism defined as serum thyroid stimulating hormone (TSH) higher than 5 mIU/mL while receiving appropriate thyroid hormone therapy Adequate thyroid-stimulating hormone (TSH) suppression (< 0.5 mIU/L) Normal thyroid function (thyroid stimulating hormone [TSH] and free thyroxine [T4]) (clinically euthyroid patients are acceptable) Thyroid stimulating hormone (TSH) within normal limits (WNL) (patients may be on thyroid hormone replacement) Thyroid-stimulating hormone (TSH) within normal limits (WNL) \r\n* Supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible Patients must have normal baseline thyroid function tests (thyroid stimulating hormone [TSH], triiodothyronine [T3], thyroxine [T4]); a history of hypothyroidism and/or hyperthyroidism is allowed, as long as the patient has stable well-controlled thyroid function for a minimum of 2 months Abnormal thyroid-stimulating hormone (TSH) Thyroid stimulating hormone (TSH) < 10 mIU/L History of uncontrolled hypothyroidism as evidenced by thyroid test (thyroid-stimulating hormone [TSH]) within the last month, hypercalcemia or hyperglycemia (within the last 15 days) Thyroid-stimulating hormone (TSH) within normal institutional limits Thyroid stimulating hormone (TSH) =< 1.5 times ULN, within 30 days prior to enrollment Thyroid stimulating hormone (TSH) 0.27 - 4.20 ulU/mL Hyperthyroidism (thyroid-stimulating hormone [TSH] < 0.4 mIU/L and free T4 > 1.6 ng/dL) Thyroid-stimulating hormone (TSH) ? 13 micro international units/mL Thyroid-stimulating hormone (TSH) < 13 micro international units/mL Prior bovine thyroid stimulating hormone (TSH) use Thyroid-stimulating hormone (TSH) < 13 micro International units/mL Thyroid-stimulating hormone (TSH) =< 10 micro international units/mL Thyroid-stimulating hormone (TSH) < 13 micro international units/mL Participants must be receiving thyroxine suppression therapy and thyroid stimulating hormone (TSH) should not be elevated (TSH should be ?5.50 mcIU/ML). When tolerated by the participant, thyroxine dose should be changed to achieve TSH suppression (TSH <0.50 mcIU/ML) and this dose may be changed concurrently upon starting study drug treatment. Patients must have histologically or cytologically diagnosed advanced anaplastic thyroid cancer (ATC) Group 4: Anaplastic thyroid cancer Patients with active, known or suspected autoimmune disease except for autoimmune thyroiditis or vitiligo. Thyroiditis patients must be asymptomatic, on adequate thyroid replacement and have normal thyroid function tests. Patients who require thyroid replacement therapy are not eligible if they have not been receiving a stable replacement dose for at least 4 weeks prior to study enrollment Phase 2 Cohort C (thyroid cancer): Subjects must have a histologically or cytologically confirmed diagnosis of metastatic thyroid cancer (stage IV) that carries either a RET rearrangement or activating RET mutation, as determined by FISH, RT-PCR, or NGS via a CLIA-certified LDT Normal thyroid and pituitary functions Uncontrolled thyroid disease or cystic fibrosis. Thyroid function tests must be consistent with stable thyroid function; patients on a stable dose of thyroid replacement therapy for a suggested minimum of 3 weeks before cycle 1, day 1 are eligible For papillary thyroid carcinoma patients: Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone may be eligible for this study. Patients who require thyroid replacement therapy are not eligible if they have not been receiving a stable replacement dose for at least 4 weeks prior to study enrollment. Eligible patients must have histopathologically confirmed Hürthle cell thyroid cancer by central review Thyroid Carcinoma Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone are eligible For the Phase I part of the study, eligible patients must have incurable poorly differentiated thyroid cancer; OR anaplastic thyroid cancer; OR radioiodine refractory differentiated thyroid cancer that is refractory to a tyrosine kinase inhibitor (TKI); OR patients who cannot tolerate a TKI are also eligible; histological confirmation of poorly differentiated, undifferentiated or anaplastic histology is required for untreated cases, but is not required for the refractory cases Patients with a history of autoimmune hypothyroidism (such as atrophic thyroiditis) on a stable dose of thyroid replacement hormone may be eligible. Pre-existing thyroid abnormality with thyroid function that cannot be maintained in the normal range with medication Baseline abnormal thyroid function tests Adequately controlled thyroid function, with no symptoms of thyroid dysfunction; patients can be on thyroid hormone replacement medication; asymptomatic patients with elevated thyroid-stimulating hormone (TSH) with normal T4/T3 are allowed to enroll, and recommended to follow with routine thyroid function test especially if they are randomized to cediranib/olaparib arm Metastatic, radioactive iodine (RAI) refractory, differentiated thyroid cancer (including papillary and follicular thyroid cancer and their sub-types such as Hurthle cell thyroid cancer as well as poorly differentiated thyroid cancer); RAI refractoriness is defined as absence of uptake of RAI on either a low dose diagnostic test or a post-treatment RAI scan in measurable lesions or radiographic progression of disease within 12 months of the last course of RAI treatment despite the recorded uptake of RAI with that previous therapy or having a cumulative lifetime administered dose of greater than 600 mCi. Diagnosis of recurrent and/or metastatic thyroid cancer Pathologically (histologically or cytologically) proven diagnosis of anaplastic thyroid cancer (a diagnosis that is noted to be consistent with anaplastic thyroid cancer is acceptable) Patients must have histologically confirmed medullary thyroid cancer by the laboratory of pathology or a pathology report and history consistent with medullary thyroid cancer; it is not uncommon for a secondary, minor pathologic focus of another form of thyroid cancer to be coincidentally found in 15-20% of patients with medullary thyroid cancer; in such cases, eligibility is based on the discretion of the investigator Adequately controlled thyroid function, with no symptoms of thyroid dysfunction; elevated thyroid stimulating hormone (TSH) with normal T3 and T4 are allowed; patients on thyroid replacement therapy are allowed Patients with thyroid dysfunction if not adequately controlled Patients with a recent history (within last 5 years) of autoimmune disease or inflammatory diseases will be excluded; exceptions will be allowed for vitiligo and hypothyroidism that has been stable on thyroid replacement medications for > 6 weeks Differentiated thyroid cancer Planned total, near-total or completion thyroidectomy requiring lifelong thyroid hormone replacement Undifferentiated, anaplastic or medullary thyroid cancer Clinically euthyroid; Note: patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism Histologically or cytologically confirmed diagnosis of anaplastic thyroid cancer or undifferentiated thyroid cancer; a diagnosis of possible anaplastic thyroid cancer (ATC)/undifferentiated thyroid cancer (UTC) will be allowed if the clinical presentation is consistent with anaplastic or undifferentiated thyroid cancer Patients with active, known, or suspected autoimmune disease\r\n* Participants with well-controlled asthma and/or mild allergic rhinitis (seasonal allergies) are eligible\r\n* Participants with the following disease conditions are also eligible: \r\n** Vitiligo,\r\n** Type 1 diabetes mellitus\r\n** Residual hypothyroidism due to autoimmune condition only requiring hormone replacement\r\n** Euthyroid participants with a history of Grave’s disease (participants suspected autoimmune thyroid disorders must be negative for thyroglobulin and thyroid peroxidase antibodies and thyroid stimulating immunoglobulin prior to first dose of study drug)\r\n** For patients with ITP (idiopathic thrombocytopenic purpura) or AIHA (autoimmune hemolytic anemia), a case by case discussion with study principal investigator (PI) may be considered Clinically euthyroid; note: patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism Subject has uncontrolled thyroid dysfunction Cohort 2A: Differentiated Thyroid Cancer (DTC) with one of the following histological subtypes: i) Papillary thyroid cancer (PTC). i.a) Follicular variant. i.b) Other 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, or poorly differentiated carcinomas). ii) Follicular thyroid cancer (FTC). ii.a) Hurthle cell. ii.b) Clear cell. ii.c) Insular. c) Cohort 2B, 3A, and 3B: Relapsed or refractory osteosarcoma. Patients must have histologically or cytologically confirmed at least 1 thyroid nodule that is >= 1 cm but =< 4 cm measured in greatest dimension and confirmed by the Laboratory of Pathology, National Cancer Institute (NCI) or confirmed by the pathology laboratory of enrolling institution:\r\n* Indeterminate thyroid biopsy per Bethesda System for reporting thyroid cytopathology with BRAF V600E mutation or RET/positron emission tomography (PET) rearrangement\r\n* Cytologically or histologically suspicious or confirmed PTC per Bethesda System for reporting thyroid cytopathology Patients who have had previous thyroid surgery Patients must have histologically or cytologically confirmed thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants); confirmation of thyroid carcinoma will be done at Memorial Sloan-Kettering (MSK) Previous thyroid surgery No uncontrolled thyroid disease (e.g. hyperthyroid/hypothyroidism) Diagnosis of recurrent and/or metastatic thyroid cancer Histological or cytological confirmation of thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants); NOTE: medullary and anaplastic thyroid cancers are excluded; Hurthle cell carcinomas are excluded (defined as having an invasive tumor composed of > 75% oncocytic [Hurthle] cells lacking the nuclear features of papillary carcinoma, tumor necrosis, and marked mitotic activity); patients with oncocytic (Hurthle cell) variants of papillary thyroid carcinoma (defined as a tumor composed of a majority of oncocytic [Hurthle] cells having the nuclear features of papillary carcinoma) are eligible to participate Thyroid function is within normal limits Patients with medullary thyroid cancer Confirmed diagnosis of differentiated thyroid cancer (follicular or papillary thyroid cancer and their variants) Adequately controlled thyroid function, with no symptoms of thyroid dysfunction; patients can be on thyroid hormone replacement medication; asymptomatic patients with elevated thyroid stimulating hormone (TSH) with normal T4/T3 are allowed to enroll, and recommended to follow with routine thyroid function test Patients with thyroid disease should be excluded unless euthyroid on suppressive or replacement therapy Histological confirmation of, or cytology reported and confirmed, anaplastic thyroid cancer in thyroid mass and/or regional lymph nodes\r\n* NOTE: A diagnosis reported as “poorly differentiated carcinoma consistent with anaplastic thyroid cancer” will be accepted Patients who require thyroid replacement therapy are not eligible if they have not been receiving a stable replacement dose for at least 28 days prior to study enrollment; patients who enter the study on thyroid replacement should have their medication adjusted to maintain thyroid-stimulating hormone (TSH) in the normal range Clinically euthyroid; note: patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism T4 within normal limits; if abnormal and patient is receiving thyroid replacement therapy, the thyroid medication may be adjusted and the T4 may be re-tested Diagnosis: Patients must have histologically confirmed medullary thyroid cancer by the Laboratory of Pathology or a pathology report and history consistent with medullary thyroid cancer; it is not uncommon for a secondary, minor pathologic focus of another form of thyroid cancer to be coincidentally found in 15-20% of patients with medullary thyroid cancer; in such cases, eligibility is based on the discretion of the investigator Pathologically (histologically or cytologically) proven diagnosis of anaplastic thyroid cancer (a diagnosis that is noted to be “consistent with anaplastic thyroid cancer” with the presence of a thyroid mass is acceptable)\r\n* Note: tissue collection for central review is mandatory, but central review is not required for eligibility; treatment will be started prior to central review Patient must not have pre-existing thyroid abnormality with thyroid stimulating hormone that cannot be maintained at less than or within the normal range with medication Thyroid dysfunction not adequately controlled Thyroid dysfunction not adequately controlled An incidental focus of medullary thyroid cancer (MTC), DTC, and/or poorly differentiated thyroid cancer in a participant with ATC is allowed. Symptomatic thyroid disease, unless negative for thyroid antibodies (TSH receptor, TPO, thyroglobulin). Pre-existing thyroid abnormality with thyroid function that cannot be maintained in the normal range with medication Uncontrolled thyroid dysfunction Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone may be eligible. Adequately controlled thyroid function, with no symptoms of thyroid dysfunction Treated medullary or papillary thyroid cancer Patients with a pre-existing thyroid abnormality who are unable to maintain thyroid function in the normal range with medication are ineligible; patients with a history of hypothyroidism are eligible provided they are currently euthyroid Any uncontrolled thyroid disease Evidence of thyroid or soft tissue invasion (stage cT4) Pathologically confirmed diagnosis of non-anaplastic non-medullary thyroid cancer that is either grossly recurrent after surgery or unresectable with or without metastatic disease Inclusion Criteria:\n\n Differentiated thyroid cancer Tumor >4 cm, or Gross extra-thyroid extension, or 1 lymph\n node >1 cm, or 5 or more lymph nodes of any size Previous thyroidectomy Must be able to\n receive radioactive iodine therapy Must be able to receive Thyroid Stimulating Hormone\n suppression\n\n Exclusion criteria:\n\n Metastaic disease Anaplastic thyroid cancer, medullary thyroid cancer or Hurthle cell\n carcinoma Presence of anti-Tg antibodies Previous treatment with any radiation Unresolved\n toxicity ? common terminology criteria for adverse event Grade 2 Patients must have histologically or cytologically confirmed papillary thyroid cancer, follicular thyroid cancer (tall cell variant, insular thyroid cancer, follicular variant of papillary thyroid cancers, poorly differentiated thyroid cancer or any of the above mixed histology will be allowed); patients with anaplastic thyroid cancer are excluded Patients with a pre-existing thyroid abnormality who are unable to maintain thyroid function in the normal range with medication are ineligible Clinically euthyroid; Note: Patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism History of autoimmune disease: patients with vitiligo or endocrine disease controlled by replacement therapy including, diabetes, thyroid and adrenal disease may be enrolled Histologic or cytologic confirmation of thyroid cancer (papillary, follicular, medullary); histologic variants such as Hurthle and tall cell variants are allowed Prior treatment with not more than 1 systemic agent (including chemotherapy or biologic agent e.g. vandetanib for patients with medullary thyroid cancer) Clinically euthyroid; Note: patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism Histopathologic confirmation of anaplastic thyroid cancer (or histopathologic report consistent with anaplastic thyroid cancer) at Memorial Sloan Kettering Cancer Center with clinical evidence of metastatic disease not curable by either surgery or radiation therapy Patients with a history of autoimmune hypothyroidism AND without normal thyroid hormone levels on a stable dose of thyroid replacement hormone Patients with well differentiated thyroid cancer are eligible for protocol as follows: For anaplastic thyroid cancer (ATC), patients must have histologically or cytologically diagnosed advanced ATC. Thyroid dysfunction not responsive to therapy Subjects known to have an uncontrolled thyroid disorder. Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone may be eligible. Uncontrolled thyroid disease Patients on active treatment for hypo- or hyperthyroidism that is uncontrolled, requiring new medication, treatment intervention (such as thyroid ablation or surgery), or a > 30% medication dose adjustment within the previous 3 months; patients on stable doses of thyroid hormone replacement for hypothyroidism or thyroid suppression for hyperthyroidism can participate localized thyroid cancer SUNITINIB MALATE ARM: Patients with a pre-existing thyroid abnormality who are unable to maintain thyroid function in the normal range with medication are ineligible The subject has concurrent uncompensated hypothyroidism or thyroid dysfunction within 10 days before the first dose of study treatment 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) 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 Patients with thyroid disease should be excluded unless their T4 is normal or they are on replacement therapy Therapy with rosiglitazone (Avandia) or pioglitazone (Actos) at any time since the diagnosis of thyroid cancer Thyroid function abnormality ? Grade 2 Uncontrolled thyroid disease or cystic fibrosis Patients with a pre-existing thyroid abnormality who are unable to maintain thyroid function in the normal range with medication are ineligible treated medullary or papillary thyroid cancer Patients must not have active or history of clinically significant autoimmune disease, defined as requiring systemic therapy; type II diabetes, vitiligo, stable hypothyroidism, and thyroid disease well controlled with thyroid replacement will not be considered exclusion criteria Clinically euthyroid; Note: patients are permitted to receive thyroid hormone supplements to treat underlying hypothyroidism Patients who are not biochemically euthyroid; patients with known history of hypothyroidism are eligible if they are on adequate and stable replacement thyroid hormone therapy for at least 3 months Subjects must be diagnosed with differentiated thyroid cancer Patient must not have any uncontrolled thyroid disease Use of thyroid medication at the time of screening Participants with thyroid carcinoma or thyroid disease for whom systemic radioactive iodine therapy is part of planned diagnostic work-up or treatment within 2 months following the contrast mammogram study. Participants with thyroid carcinoma or thyroid disease for whom systemic radioactive iodine therapy is part of planned diagnostic work-up or treatment within 2 months following the contrast mammogram study History of known thyroid disease History of known thyroid disease History of thyroid cancer Patient must be status post near total thyroidectomy for differentiated thyroid cancer without known distant metastases and who are planning to undergo routine remnant thyroid tissue ablation with I-131 Patients must qualify for thyroid ablation with I-131 Known metastatic thyroid cancer Participants with thyroid carcinoma or thyroid disease for whom systemic radioactive iodine therapy is part of planned diagnostic work-up or treatment within 2 months following the contrast mammogram study Participants with actively treated thyroid disease who still have a portion or all of their thyroid gland Participants with a history of Graves disease, goiter, thyroid nodules, Hashimoto’s thyroiditis, post-partum thyroiditis, type 2 amiodarone-induced thyrotoxicosis, or subacute thyroiditis who still have a portion or all of their thyroid gland Abnormal thyroid function, such as untreated clinical diagnosis of hypothyroidism, hyperthyroidism, or other thyroid disease Patient with thyroid cancers Biopsy proven papillary thyroid cancer (regardless of genotype and including all subtypes such as follicular or mixed papillary follicular) or suspicious for thyroid cancer Thyroidectomy or lobectomy planned as definitive treatment for thyroid cancer or patients on active surveillance management approach Subjects must have either primary or recurrent papillary thyroid cancer (PTC) or medullary thyroid carcinoma (MTC) with nodal metastases detected through physical examination and/or standard imaging techniques Locally recurrent and unresectable and/or distant metastatic differentiated thyroid cancer (DTC), histologically or cytologically confirmed; the diagnosis of DTC includes the following subtypes: papillary thyroid cancer (PTC) (including but not limited to variants such as follicular variant, tall cell, columnar cell, Hurthle cell variant of papillary carcinoma, and poorly differentiated), follicular thyroid cancer (FTC), including insular variant, Hurthle cell carcinoma and poorly differentiated thyroid cancer All nasopharyngeal, paranasal sinus, salivary cancer, and thyroid malignancies Preoperative fine needle aspiration cytology classified as papillary thyroid carcinoma or suspicious for papillary thyroid carcinoma. Patients with a history of prior thyroid/parathyroid surgery will be excluded from the study. Use of thyroid replacement medication (Synthroid or similar) for < 1 year Patients who have already undergone any thyroid surgery will be excluded History of autoimmune disorders, including rheumatic diseases and thyroid disorders. Exception: As with bone marrow donations, donors with a history of thyroid disease who have undergone successful therapy may be suitable. Individual adult patient with current or previous known or suspected thyroid cancer or nodule(s) if they come from a family with a high suspicion of hereditary cancer Individuals from families with a high suspicion of hereditary thyroid cancer:\r\n* Families with a current or previous diagnosis of a thyroid cancer/nodule occurring in childhood (< 18 years old)\r\n* Families with a high suspicion of hereditary thyroid cancer/nodules other than above to include:\r\n** Families with thyroid cancer in multiple individuals\r\n** Families with thyroid cancer and a known genetic syndrome\r\n** Families with thyroid cancer and a suspected genetic syndrome (e.g. multiple childhood cancers in the family, multiple primary cancers, multiple endocrinopathies, etc.) Papillary thyroid cancer (PTC) Follicular thyroid cancer (FTC) Anaplastic or medullary carcinoma of the thyroid.