This research study is investigating Binimetinib and Belinostat in participants with metastatic uveal melanoma. The research study will test the study drugs to see if the combination of binimetinib and belinostat can make tumors shrink or stop growing.

Inclusion Criteria:

• Provision of signed and dated informed consent form.
• Male or female, aged >/= 18 years old
• Life expectancy of greater than 3 months in the opinion of the investigator
• Must have metastatic uveal melanoma, either initial presentation or recurrent, that is histologically diagnosed
• Participant must have ECOG performance status of 0-1
• Participant must have measurable disease, according to RECIST version 1.1
• Participants must have normal organ and marrow function as defined below:
• Leukocytes >3,000/mcL
• Absolute neutrophil count >1,500/mcL
• Platelets >100,000/mcL
• Total bilirubin within 1.5 x institutional upper limit of normal
• AST(SGOT)/ALT(SGPT) <2.5 X institutional upper limit of normal
• Creatinine within 1.5 x institutional upper limit of normal OR creatinine clearance >60 mL/min/1.73 m2 for participants with creatinine levels above institutional normal
• An echocardiogram should be performed at baseline in all patients. Ejection fraction (EF) from baseline echocardiogram must be within the institutional limits of normal as determined by the reading cardiologist. The left ventricular ejection fraction (LVEF) must be ≥50%
• Participants on full-dose anticoagulants (e.g., warfarin) with PT INR >1.5 are eligible provided that both of the following criteria are met:
• a) The participant has an in-range INR (usually between 2 and 3) on a stable dose of oral anticoagulant or on a stable dose of low molecular weight heparin
• b) The participant has no active bleeding or pathological condition that carries a high risk of bleeding (e.g., tumor involving major vessels or known varices)
• A participant may be treatment naïve. However, prior systemic treatments for metastatic uveal melanoma are allowed. There is no limit on the number of prior regimens for metastatic uveal melanoma. However, no prior therapy with a MEK inhibitor or an HDAC inhibitor
• Participant must be free of active brain metastasis by contrast-enhanced CT/MRI scans within 4 weeks prior to enrollment. If known to have prior brain metastases, these must have been adequately managed with standard of care radiation therapy, stereotactic radiosurgery or surgery prior to registration on the study
• For females of reproductive potential: use of highly effective contraception for at least 1 month prior to screening and agreement to use such a method during study participation and for an additional 6 months after the end of any study drug administration
• For males of reproductive potential and/or with female partners of reproductive potential: use of condoms or other methods to ensure effective contraception with partner for 3 months after the last dose of any study drug.

Exclusion Criteria:

• Pregnancy or lactation
• Treatment with another investigational drug or other systemic intervention for uveal melanoma within 4 weeks of initiation of study drugs. Participants must not have radiotherapy within the preceding 4 weeks. Participants must have recovered from adverse events due to agents administered more than 4 weeks earlier
• Participants must be at least 4 weeks from major surgery and have fully recovered from any effects of surgery and be free of significant detectable infection
• Participants must not have other significant medical, surgical, or psychiatric conditions or require any medication or treatment that in the opinion of the investigator may interfere with compliance, make the administration of the study drugs hazardous or obscure the interpretation of AEs
• Participants must not have an active infection requiring current treatment with parenteral antibiotics
• Cardiac: No evidence of congestive heart failure, symptoms of coronary artery disease, myocardial infarction less than 6 months prior to entry, serious cardiac arrhythmias, or unstable angina
• CNS: No history of cerebrovascular accident or transient ischemic attacks within the past 6 months
• Serious or non-healing wound, ulcer, or bone fracture
• History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 4 weeks of initiating study treatment
• Participants with clinically significant cardiovascular or cerebrovascular disease:
• History of cerebrovascular accident or transient ischemic attack within past 6 months
• Uncontrolled hypertension, defined as blood pressure >150/100 mm Hg or systolic BP >180 mm Hg if diastolic blood pressure <90 mm Hg, on at least 2 repeated determinations on separate days within past 3 months
• Myocardial infarction, CABG or unstable angina within the past 6 Months
• New York Heart Association grade III or greater congestive heart failure (Appendix E), serious cardiac arrhythmia requiring medication, unstable angina pectoris within past 6 months
• Clinically significant peripheral vascular disease within past 6 months
• Pulmonary embolism, DVT, or other thromboembolic event within past 6 months
• PT INR >1.5 unless the patient is on full-dose warfarin
• Participants who have other current malignancies are not eligible. Participants with other malignancies are eligible if they have been continuously disease free for > 3 years prior to the time of study registration (enrollment). Participants with prior history at any time of any in situ cancer, lobular carcinoma of the breast in situ, cervical cancer in situ, atypical melanocytic hyperplasia or melanoma in situ are eligible. Participants with prior history of basal or squamous skin cancer are eligible
• History of retinal vein occlusion, uveitis refractory to ocular therapy, and symptomatic serous retinopathy or retinal pigment epithelial detachments
• Active requirement for or a history of corticosteroid systemic therapy in order to treat Interstitial lung disease (ILD) or pneumonitis
• The presence of a disorder that may impact absorption of study drugs, such as inability to take oral medication, requirement for IV alimentation, prior gastric resection, treatment for active peptic ulcer confirmed by endoscopy within the past 3 months, active GI bleed, GI malabsorption syndrome