A UGT1A1 Genotype-Directed Study of Belinostat Pharmacokinetics and Toxicity

-INCLUSION CRITERIA: 1. Participants must have histologically confirmed diagnosis of Extrapulmonary High-Grade Neuroendocrine Neoplasms (HGNENs) for which there is no known standard therapy capable of extending life expectancy. 2. Age \>= 18 years. 3. Participants with neuroendocrine prostate cancer may continue ongoing LHRH agonist therapy. 4. Participants with bone metastases or hypercalcemia who began intravenous bisphosphonate treatment prior to study entry may continue this treatment while on study. 5. Evaluable (measurable or non-measurable) disease, per RECIST 1.1. 6. ECOG performance status \<=2 at screening 7. Participants must have adequate organ and marrow function as defined below: * Leukocytes \>=3,000/mcL * Hemoglobin \>= 10 g/dL * Absolute neutrophil count (ANC) \>=1,500/mcL * Platelets \>=100,000/mcL * Aspartate aminotransferase (AST) or serum glutamic-oxaloacetic transaminase (SGOT) / Alanine aminotransferase (ALT) or serum glutamic-pyruvic transaminase (SGPT): \<=3 X institutional upper limit of normal * Total bilirubin \<= 1.5 x institutional upper limit of normal (ULN). NOTE: In participants with Gilbert s syndrome, a total bilirubin \<= 3.0 X ULN is allowed * Serum Creatinine \<= 1.5 X institutional ULN OR * An estimated Creatinine clearance (CrCL) \>=60 mL/min/1.73 m\^2 based on the Cockcroft Gault equation * Prothrombin time (PT) / International normalized ratio (INR) and Partial thromboplastin time (PTT) \<= 1 X institutional ULN 8. Hepatitis B virus (HBV)-infected participants can be enrolled if HBV DNA is undetectable. Hepatitis C virus (HCV)-infected participants can be enrolled if HCV RNA level is undetectable 9. Women of child-bearing potential (WOCBP) must agree to use effective contraception (hormonal, intrauterine device (IUD), tube ligation, a partner has had a previous vasectomy, abstinence) prior to study entry, during the study, and for 14 months for women after the last dose of the study drug(s). Men with partners of childbearing potential must agree to use effective contraception (abstinence, condoms, previous vasectomy) or request partners to use effective contraception (per above) during the study and for 11 months after the last dose of study therapy. 10. Breastfeeding participants must be willing to discontinue breastfeeding starting with prior to study entry, during the study, and for 3 months after the last dose of the study drug(s). 11. Willing to comply with study procedures and follow-up. 12. Participants must be able to understand and be willing to sign a written informed consent document. EXCLUSION CRITERIA: 1. Participants with prior investigational drug, chemotherapy, immunotherapy or any prior radiotherapy (except for palliative bone directed therapy) within the past 14 days prior to the first drug administration. Additionally, FDA-approved hormonal therapy for the treatment or prevention of other malignancies (e.g., breast cancer, prostate cancer) may be continued where in the opinion of the investigator stopping such therapies may increase the risk of disease progression. Potential drug-drug interactions with the hormonal agent will be assessed by the investigator prior to enrollment. 2. History of allergic reactions attributed to compounds of similar chemical or biologic composition to belinostat, cisplatin, etoposide or other agents used in study. Participants with a history of allergic reactions to medications containing polysorbate 80 will be evaluated on a case-by-case basis. 3. Participants with treated brain metastases are not eligible except if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. 4. Participants who have not recovered (CTCAE \<= grade 1) from non heme adverse events due to prior treatments, except for alopecia, or stable grade 2 tinnitus (not interfering with ADL's) or baseline hearing loss by audiometry or stable grade 2 sensory neuropathy (moderate symptoms; limiting instrumental ADL) without pain or motor component, and not interfering with ADL's. 5. Participants taking strong UGT1A1 inhibitors or CYP3A4 inhibitors or inducers must discontinue their use a minimum of 5 half-lives prior to starting treatment on this trial 6. Participants with platinum-refractory disease. 7. Participants who have had another histone deacetylase inhibitor (e.g., valproic acid, vorinostat) for at least 2 weeks prior to enrollment. 8. Participants who have had radiation to the pelvis or other bone marrow-bearing sites will be considered on a case-by-case basis and may be excluded if the bone marrow reserve is not considered adequate (\>25% of bone marrow). 9. Pregnancy (confirmed with beta-Human chorionic gonadotropin (HCG) serum or urine pregnancy test performed in WOCBP at screening) 10. Significant cardiovascular disease (New York Heart Association Class III or IV cardiac disease), myocardial infarction within the past 6 months, unstable angina, unstable arrhythmia, or a need for anti-arrhythmic therapy (use of medication to control heart rate in participants with atrial fibrillation is allowed, if stable medication for at least last month prior to enrollment and medication not listed as causing Torsade de Points), or evidence of acute ischemia on ECG. 11. Baseline prolongation of QT/QTc interval, i.e., defined as an average QTc interval \> 450 msec calculated using the Fridericia formula for QT correction; Long QT Syndrome; or the required use of concomitant medication that may cause Torsade de Pointes. 12. Participants with HIV infection if CD4 count \<200 cells per cubic millimeter before treatment initiation 13. Uncontrolled intercurrent illness that would limit compliance with study requirements.