5-aza-4'-Thio-2'-Deoxycytidine (Aza-TdC) in People With Advanced Solid Tumors

* INCLUSION CRITERIA: * Patients must have histologically documented solid tumors whose disease has progressed on standard therapy or for which there is no available standard therapy. * Age \>=18 years of age. * ECOG performance status \<= 2. * Patients must have normal organ and marrow function as defined below: * absolute neutrophil count \>= 1,500/mcL * platelets \>=100,000/mcL * total bilirubin \<=1.5 X institutional upper limit of normal (\<=3 x upper limit of normal in the presence of documented Gilbert s syndrome) * AST(SGOT)/ALT(SGPT) \<=3 X institutional upper limit of normal OR * AST(SGOT)/ALT(SGPT) \<=5 X institutional upper limit of normal for patients with liver metastases * creatinine \<=1.5X institutional upper limit of normal OR * creatinine clearance \>=60 mL/min/1.73 m\^2 for patients with creatinine levels above 1.5X institutional normal * Because nucleoside analogs are known to be teratogenic, women of child-bearing potential and men must agree to use two forms of contraception (hormonal or barrier method of birth control; abstinence; sterilization) prior to study entry, for the duration of study participation, and for 3 months after completing 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. Men treated or enrolled on this protocol must also agree to use two forms of contraception prior to the study, for the duration of study participation, and for 3 months after completion of administration of Aza-TdC. * Patients must have completed any chemotherapy or biologic therapy \>= 4 weeks or 5 half-lives (whichever is shorter) (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. Patients must be \>= 2 weeks since any prior palliative radiation or cyberknife therapy. Patients must have recovered to grade 1 from prior toxicity or adverse events. Patients on study may be eligible for palliative radiotherapy to non-targeted lesions after 2 cycles of therapy at the PI's discretion. Patients with bone metastases or hypercalcemia on intravenous bisphosphonate treatment prior to study entry may continue this treatment. * Ability to understand and the willingness to sign a written informed consent document. * Willingness to provide blood and urine samples for research purposes. * Ability to swallow pills/capsules. * Left ventricular ejection fraction greater than 45% or the institutional lower limit of normal by either ECHO or MUGA at entry. * For patients enrolled on the expansion cohort, patients must have tumor amenable to biopsy (excisional or incision biopsies of skin or H \& N lesions under visualization) and willingness to undergo tumor biopsies. EXCLUSION CRITERIA: * Patients who are receiving any other investigational agents. * Pregnant women and women who are breastfeeding are excluded from this study. * Patients with clinically significant illnesses which would compromise participation in the study, including, but not limited to active or uncontrolled infection, immune deficiencies, known HIV infection requiring protease inhibitor therapy, known Hepatitis B, known Hepatitis C, uncontrolled diabetes, uncontrolled hypertension, symptomatic congestive heart failure, unstable angina pectoris, myocardial infarction within the past 6 months, uncontrolled cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Patients with known primary central nervous system (CNS) malignancy or symptomatic CNS metastases are excluded, with the following exceptions: * Patients with asymptomatic untreated CNS disease may be enrolled, provided all of the following criteria are met: * Evaluable or measurable disease outside the CNS * No metastases to brain stem, midbrain, pons, medulla, or cerebellum * No history of intracranial hemorrhage or spinal cord hemorrhage * No ongoing requirement for dexamethasone for CNS disease; patients on a stable dose of anticonvulsants are permitted. * No neurosurgical resection or brain biopsy within 28 days prior to Cycle 1, Day 1 * Patients with treated CNS metastases may be enrolled, provided all the criteria listed above are met as well as the following: * No stereotactic radiation or whole-brain radiation within 14 days prior to Cycle 1, Day 1 * Screening CNS radiographic study 2 weeks from completion of radiotherapy and \>=1 week from discontinuation of corticosteroids. The presence of new CNS mets will not exclude the patient but provide a baseline. If the irradiated lesion showed increased edema or growth, patient may be enrolled if asymptomatic but a repeat MRI should be done within the next 2-4 weeks for follow up. * Malabsorption syndrome or other conditions that would interfere with intestinal absorption.