Inclusion Criteria:

        For dose escalation cohorts only:

        - Pathologically confirmed breast cancer with the following features:

          -  Measurable disease by RECIST 1.1;

          -  ER and PR ≤ 1% by immunohistochemistry;

          -  Her-2/neu negative (0 or 1+ by immunohistochemistry OR not-amplified by CAP/ASCO
             standards);

          -  Metastatic or unresectable and locally advanced and not amenable to treatment with
             curative intent, in the opinion of the enrolling investigator.

        OR --Pathologically confirmed serous ovarian cancer that is recurrent and is unresectable,
        in the opinion of the enrolling investigator.

        For dose expansion cohort only:

        - Pathologically confirmed breast cancer with the following features:

          -  Measurable disease by RECIST 1.1;

          -  ER and PR ≤ 1% by immunohistochemistry;

          -  Her-2/neu negative (0 or 1+ by immunohistochemistry OR not-amplified by CAP/ASCO
             standards);

          -  Metastatic or unresectable and locally advanced and not amenable to treatment with
             curative intent, in the opinion of the enrolling investigator.

        For all patients:

          -  Age ≥ 18.

          -  ECOG performance Status ≤ 2.

          -  Able to swallow pills.

          -  Adequate organ function as defined as:

               -  Hematologic:

                    -  ANC > 1,500/mm3

                    -  Platelets > 100,000/mm3

                    -  Hemoglobin > 9g/dL

               -  Hepatic:

                    -  Serum bilirubin levels ≤1.5 mg/dL. Higher levels are acceptable if these can
                       be attributed to active hemolysis or ineffective erythropoiesis. Bilirubin
                       above 1.5mg/dL due to Gilbert's is still excluded.

                    -  Serum aspartate transaminase (AST) and serum alanine transaminase (ALT) ≤
                       2.5 X upper limit of normal.

                    -  AST(SGOT)/ALT(SGPT) ≤ 5 × institutional ULN for patients with liver
                       metastasis.

                    -  Alkaline phosphatase < 2.5 X upper limit of normal, unless bone metastasis
                       is present in the absence of liver metastasis

               -  Renal:

                    -  Serum creatinine levels ≤1.5 mg/dL

                    -  Patient must have the following laboratory values within normal limits or
                       corrected to within normal limits with supplements before the first dose of
                       study medication:

                         -  Potassium

                         -  Magnesium

                         -  Total Calcium (corrected for serum albumin)

               -  Coagulation ---INR ≤1.5 (unless the patient is receiving anticoagulants and the
                  INR is within the therapeutic range of intended use for that anticoagulant within
                  7 days prior to the first dose of study drug)

          -  Presence of ≥ 1 metastatic sites of disease that can be safely accessed for biopsy and
             patient willingness to undergo fresh tissue biopsies of up to 3 lesions.

          -  Negative serum or urine pregnancy test at screening for women of childbearing
             potential.

          -  Agrees to continue use of approved birth control for at least 6 months after receiving
             the last dose of study drugs. See section 7.3 for list of approved birth control
             methods.

          -  Able to provide informed consent and have signed an approved consent form that
             conforms to federal and institutional guidelines.

        Exclusion Criteria:

          -  Previous use of CDK 4/6 or HDAC inhibitors for cancer treatment

          -  Major surgery, radiotherapy, anticancer therapy, or investigational agents ≤ 4 weeks
             of treatment day 1 or ≤5 half-lives, whichever is shorter.

          -  Patients with new or progressive brain metastases (active brain metastases) or
             leptomeningeal disease unless determined by the treating physician that immediate CNS
             specific treatment is not required and is unlikely to be required during the first
             cycle of therapy.

          -  Medical condition that in the opinion of the enrolling investigator would require the
             use of valproic acid within ≤ 5 days of the first dose of belinostat or while on
             study.

          -  Active infection requiring systemic therapy.

          -  History of allergy or hypersensitivity to belinostat, ribociclib, or their binders.

          -  Uncontrolled arrhythmia, congestive heart failure or angina. Patients who have had a
             myocardial infarction, symptomatic pericarditis, or cardiac surgery should be at least
             6 months from the event and free of active symptoms

          -  Known left ventricular ejection fraction < 50%. (Echocardiogram is not required for
             study entry)

          -  Clinically significant cardiac arrhythmias (e.g., ventricular tachycardia), complete
             left bundle branch block, high-grade AV block (e.g., bifascicular block, Mobitz type
             II and third degree AV block)

          -  Congenital long QT syndrome.

          -  Baseline QTcF>450 msec. The heart rate on the qualifying ECG must be between 50 and 90
             BPM.

          -  Concurrent use of medication known to inhibit UGT1A1. Patients currently taking these
             medications must have discontinued ≥7 days prior to treatment day 1.

          -  Concurrent use of herbal supplements, unless approved by the prinicipal investigator.
             Patients currently taking herbal supplements must have discontinued ≥ 7 days prior to
             treatment day 1.

          -  Concurrent use of medication with a known risk of inducing Torsades de Pointes (on the
             known risk list of crediblemeds.org) that cannot be discontinued or switched to a
             different medication ≥ 7 days prior to starting the study drug.

          -  Unresolved diarrhea ≥ Grade 2, per CTCAE v5.0.

          -  Use of any of the following substances ≤ 7 days prior to the start of the treatment:

               -  Known strong and moderate inducers or inhibitors of CYP3A4/5, including
                  grapefruit, grapefruit hybrids, pomelos, star-fruit, and Seville oranges.

               -  Medications that have a narrow therapeutic window and are predominantly
                  metabolized through CYP3A4/5.

          -  Patient is currently receiving warfarin or other coumarin derived anti-coagulant, for
             treatment, prophylaxis or otherwise.

             --Note: Therapy with heparin, low molecular weight heparin (LMWH), an oral factor Xa
             inhibitor, an oral direct thrombin inhibitor, or fondaparinux is allowed.

          -  Impaired GI function that may alter absorption of medicines, such as uncontrolled
             inflammatory bowel disease, uncontrolled vomiting, or major stomach or small bowel
             resection.

          -  Pregnant or breast feeding

          -  Women of child-bearing potential defined as all women physiologically capable of
             becoming pregnant or men whose female partner is of child-bearing potential, unless
             they are using highly effective methods of contraception during the study treatment
             and for 6 months after stopping the treatment. Highly effective contraception methods
             include:

               -  Total abstinence (when this is in line with the preferred and usual lifestyle of
                  the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal,
                  post-ovulation methods) and withdrawal are not acceptable methods of
                  contraception

               -  Female sterilization (have had surgical bilateral oophorectomy with or without
                  hysterectomy), total hysterectomy or tubal ligation at least 6 weeks before
                  taking study treatment. In case of oophorectomy alone, only when the reproductive
                  status of the woman has been confirmed by follow up hormone level assessment

               -  Male partner sterilization (at least 6 months prior to screening). For female
                  patients on the study, the vasectomized male partner should be the sole partner
                  for that patient and the success of the vasectomy must be medically confirmed as
                  per local practice.

               -  Placement of an intrauterine device (IUD).

               -  Use of hormonal contraception plus a barrier contraceptive.

          -  Known HIV infection with a detectable viral load within 6 months of the anticipated
             start of treatment.

        Note: Patients on effective anti-retroviral therapy with an undetectable viral load within
        6 months of the anticipated start of treatment are eligible for this trial.

          -  Known chronic hepatitis B virus (HBV) or hepatitis C virus infection with a detectable
             viral load.

             --Note: Patients with an undetectable HBV viral load on appropriate suppressive
             therapy are eligible. Patients with an undetectable HCV viral load are eligible.

          -  Malignancy other than breast carcinoma or ovarian cancer (dose escalation) anticipated
             to need systemic treatment within 1 year in the opinion of the enrolling investigator.