This is a phase 1 study in which 2 to 72 patients with advanced cancer will receive oral doses of rigosertib, a new investigational (unapproved) anti-cancer drug. The objective of the study is to determine the highest dose of drug that can be given safely. The study will start by testing a low dose. If this dose is safe, then, higher and higher doses will be tested as long as the previous lower dose was safe. Safety will be determined by looking for any side effects or unusual laboratory values. It is important to know the highest safe dose so that additional studies can be done. The drug will be given in the form of capsules twice or three times a day for 21 consecutive days of 21-day cycles.

Patients will be initially enrolled in two-patient cohorts starting with a 70 mg bid dosing.

• In the absence of drug-related grade 2 or higher toxicity in the two patients treated for an entire 21-day first cycle, the next two patients will be receiving a dose escalated by 100% from prior dose.
• If drug-related grade 2 or higher toxicity is observed in at least one of the two patients treated for a full 21-day cycle, the cohort will be expanded in order to obtain 3 evaluable (treated for an entire 21-day first cycle) patients.
• If no dose limiting toxicity (DLT) is observed in the first three patients treated for an entire 21-day first cycle, then the next three patients will be enrolled at a dose level increased by approximately 50% from prior dose.
• If one DLT is observed in the first three patients treated for an entire 21-day first cycle, then the three next patients will be enrolled at the same dose level.
• If no more than one DLT is observed in the six patients treated for an entire 21-day cycle, then the next six patients will be administered a dose level increased by approximately 25% from prior dose.
• If two or more patients in any cohort experience DLT, then the maximum tolerated dose (MTD) will have been exceeded and no further dose escalation will occur. The MTD will be established as the immediate prior dosing level
• Identical rules will be applied to all cohorts of patients recruited to the study.
• A total of up to 24 patients may be treated at the MTD dose level in order to obtain data on the onset and severity of dysuria symptoms in approximately 12 patients including about 6 patients who will be treated with oral sodium bicarbonate at the time of treatment initiation (early treatment) and ~6 patients who will be treated with oral sodium bicarbonate at the time of symptom onset (late treatment). Alternating patients on a 1-to-1 basis by enrollment date will receive oral sodium bicarbonate as 650 mg tablets given twice daily with two 8-ounce glasses of water 1 hour after rigosertib administration, either at the time of study initiation (early treatment) or at the time of urinary symptom onset (late treatment).

Three additional cohorts will be treated with multiple escalating doses administered orally three time daily (TID) (140 mg, 280 mg and 420 mg) for 21 consecutive days of a 21-day cycle using identical rules as those described above (ie, starting with 2-patient cohorts at 140 mg dosing and expanding to 3-6 patient cohorts if at least 1 Grade 2 or higher drug-related toxicity is observed in 2-patient cohorts).

Up to 12 patients may be treated at the MTD level identified with TID dosing.

Prior to escalating to the next planned dose level, a designated Cohort Review Committee (CRC) consisting of the Principal Investigators (or their representatives), the Medical Monitor and the Sponsor's Medical Representative will review all available safety data over the 21 day duration of Course 1 for the previous cohort. Additional input may be provided from the study monitors, pharmacokinetic or toxicology specialists as required. The CRC also has to ability to recommend dose de-escalation for ON 01910.Na if warranted by the observed toxicity profile.

Inclusion Criteria:

1. Histologically confirmed solid tumor (leukemias and lymphomas excluded).
2. Malignancy that is incurable and for which standard (FDA approved or established standard clinical practice), curative, or palliative measures do not exist or are no longer effective.
3. ECOG performance status 2, 1 or 0.
4. Life expectancy greater than 6 months.
5. One or more measurable lesion(s) ("target lesion[s]"), that can be accurately measured in at least 1 dimension with longest diameter equal to or greater than 20 mm using conventional techniques (computed tomography [CT] scan or magnetic resonance imaging [MRI]) or equal to or greater than 10 mm with spiral CT scan.
6. If female, has a negative screening for pregnancy. Women of child-bearing potential and men must agree to use adequate contraception prior to study entry (hormonal or barrier method of birth control; abstinence) and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately.
7. Ability to understand the nature of the study and any hazards of participating in it, to communicate satisfactorily with the investigator, and to participate in, and comply with, the requirements of the entire study.
8. Willing to adhere to the prohibitions and restrictions specified in this protocol.
9. Patient must have signed an informed consent document.

Exclusion Criteria:

1. Recent major surgery (within the past 14 days).
2. Chemotherapy or dose of other potentially myelosuppressive treatment within 3 weeks prior to entering the study (6 weeks for nitrosoureas or mitomycin C).
3. Among patients with prior doxorubicin chemotherapy, only those with no more than a total cumulative dose of 450 mg/m2 of the drug.
4. Definitive radiotherapy (over 10 fractions and maximal area of hematopoietic active bone marrow treated was greater than 25%) within 4 weeks prior to entering the study.
5. Palliative radiotherapy (10 or less fractions) within 2 weeks prior to entering the study.
6. Residual adverse events (except alopecia, stable residual neuropathy, and residual hand, foot syndrome) and ascites requiring active medical management including paracentesis, peripheral bilateral edema, hyponatremia (serum value less than 130 Meq/L) due to previously administered agents, which have not recovered at grade 1 severity level or below before study entry.
7. Receiving any other investigational agents or concurrent chemotherapy, radiotherapy, hormonal treatments, bone marrow transplantation, or immunotherapy while on study. Exceptions are long-term hormonals for prostate (e.g. goserelin) and octreotide for neuroendocrine malignancies.
8. Previous bone marrow transplant.
9. Known brain metastases, except brain metastases that have been previously removed or irradiated and currently have no clinical impact.
10. History of allergic reactions attributed to compounds of similar chemical or biologic composition to ON 01910.Na.
11. Uncontrolled intercurrent illness.
12. Hgb less than 9 gm/dL (must not require transfusional support but erythropoietin therapy is permitted).
13. WBC less than 4,000/microliter.
14. Absolute neutrophil count less than 1,500/microliter.
15. Platelets less than or equal to 100,000/microliter.
16. Total bilirubin greater than 1.5 times institutional upper normal limit.
17. AST(SGOT)/ALT(SGPT) equal to or greater than 2.5 x institutional upper normal limit. (If liver function abnormalities are due to metastatic disease, patients are eligible provided the transaminases are < 5 times institutional upper normal limit.).
18. Serum creatinine greater than 2 times upper limit of institutional normal value.
19. Pregnant and nursing women.
20. HIV-1 positive patients receiving combination anti-retroviral therapy.