Inclusion Criteria:

A subject who meets all of the following criteria may be included in the study:

• Part 1: HCC; cholangiocarcinoma; or esophageal, nasopharyngeal, or serous ovarian cancers, regardless of FGF/FGFR alteration status.
• Part 2: Subjects will be enrolled into 1 of 3 cohorts:
  • Cohort A: HCC with FGF19 amplification;
  • Cohort B: HCC without FGF19 amplification; and
  • Cohort C: cholangiocarcinoma or esophageal, nasopharyngeal or serous ovarian cancers (regardless of FGF/FGFR status), or other solid tumor malignancies with documented FGF19/FGFR4 alteration (FGF19/FGFR4 pathway activating alterations include, but are not limited to, FR4 amplification, FGFR4 activating mutations, and FGF19 amplification) based on local or central testing.
• Has progressed after prior therapy and either a) there is no further effective standard anticancer therapy available (including subject refusal) or b) subject is intolerant to standard anticancer therapy.
• Life expectancy > 12 weeks.
• ECOG performance status:
  • a. Part 1: 0-1.
  • b. Part 2: 0-2.
• Archival tumor specimen:
  • Tumor block (formalin-fixed paraffin-embedded) or approximately 20 unstained slides or willingness to undergo a pretreatment tumor biopsy to provide a tumor block or approximately 20 unstained slides.  Archival tumor biopsies are acceptable at baseline if collected within approximately 2 years from the date of signing consent.
  • Subjects with a sequencing report from the sponsor's central laboratory must have available archival tissue for confirmatory testing; if archival tissue is not available, a fresh biopsy will be required unless approved by sponsor medical monitor.
• Willingness to avoid pregnancy or fathering children based on the criteria below:
  • Woman of nonchildbearing potential (ie, surgically sterile with a hysterectomy and/orbilateral oophorectomy OR chemically sterile OR ≥ 12 months of amenorrhea.)
  • Woman of childbearing potential who has a negative serum pregnancy test at screening and before the first dose on Day 1 and who agrees to take appropriate precautions to avoid pregnancy (with around 99% certainty) from screening through 90 days after last dose of study drug. Permitted methods that are around 99% effective in preventing pregnancy (see Appendix A) should be communicated to the subject and their understanding confirmed.
  • Man who agrees to take appropriate precautions to avoid fathering children (with at least 99% certainty) from screening through 90 days after last dose of study drug. Permitted methods that are around 99% effective in preventing pregnancy (see Appendix A) should be communicated to the subject and their understanding confirmed.

Exclusion Criteria:

A subject who meets any of the following criteria will be excluded from the study:

• Treatment with other investigational study drug for any indication for any reason, or receipt of anticancer medications within 28 days before first dose of study drug (6 weeks for mitomycin-C or nitrosoureas, 7 days for tyrosine kinase inhibitors); subjects must have recovered (≤ Grade 1 or pretherapy baseline) from AEs due to previously administered therapies.
• Prior receipt of a selective FGFR4 inhibitor within the last 6 months.
• Laboratory parameters outside Protocol-defined range:
• Part 1 Dose Escalation:
  • Hemoglobin < 10.0 g/dL
  • Platelet count < 100 × 109/L
  • ANC < 1.5 × 109/L
  • Total bilirubin > ULN
  • AST or ALT > 1.5 × ULN
  • ALP > 1.5 × ULN (> 3 × ULN if related to underlying disease)
  • Creatinine clearance ≤60 mL/min based on the site's standard formula
  • Prothrombin time > ULN
  • Serum calcium outside of normal range per institution or serum albumin-correct calcium outside of the institutional normal range when serum albumin is outside of the institutional normal range
  • Serum phosphate above ULN per institution
• Part 2 Expansion:
  • Hemoglobin ≤9.0 g/dL
  • Platelet count ≤75 × 109/L
  • ANC ≤ 1.0 × 109/L
  • Total bilirubin ≥ 1.5 × ULN
  • AST or ALT ≥5 × ULN
  • ALP ≥ 2.5 × ULN (> 5 × ULN if related to underlying disease)
  • Creatinine clearance ≤40 mL/min based on the site's standard formula
  • Prothrombin time > 1.5 × ULN
  • Serum calcium outside of normal range per institution or serum albumin-correct calcium outside of the institutional normal range when serum albumin is outside of the institutional normal range
  • Serum phosphate above ULN per institution
  • Has a history or presence of an abnormal ECG that in the investigator's opinion is clinically meaningful. A screening QTc interval > 450 milliseconds, as corrected by Fridericia, is excluded.
  • Prior radiotherapy within 2 weeks of study treatment. Subjects must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. Evidence of fibrosis within a radiation field from prior radiotherapy is permitted with medical monitor approval. A 1-week washout period is permitted for palliative radiation to non–NS disease with medical monitor approval.
• History of human immunodeficiency virus infection.
• Untreated brain or CNS metastases or brain/CNS metastases that have progressed (eg, evidence of new or enlarging brain metastasis or new neurological symptoms attributable to brain/CNS metastases). Subjects with previously treated and clinically stable brain/CNS metastases and who are off all  orticosteroids for ≥4 weeks are eligible.
• Chronic or current active infectious disease requiring systemic antibiotic, antifungal, or antiviral treatment, except concomitant antiviral systemic therapy for chronic hepatitis B or C.
• Child–ugh liver function Class B or C.
• History of clinically significant or uncontrolled cardiac disease including unstable angina, acute myocardial infarction, New York Heart Association Class III or IV congestive heart failure, or arrhythmia requiring therapy.
• History of allergic reactions to INCB062079, any of the excipients of INCB062079 or similar compounds (refer to the IB).
• Unable or unwilling to swallow and retain INCB062079 tablets.
• Pregnant or nursing women or subjects expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 90 days after last dose of study drug.
• Any medical condition that would in the investigator's judgment interfere with full participation in the study, including administration of study medication and attending required study visits; pose a significant risk to the subject; or interfere with interpretation of study data.
• Concurrent anticancer therapy (eg, chemotherapy, radiation therapy, surgery, immunotherapy, biologic therapy, hormonal therapy, investigational therapy, or tumor embolization).
• Known additional malignancy that is progressing or requires active treatment.  Exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy.
• History of calcium and phosphate hemostasis disorder or systemic mineral imbalance with ectopic calcification of soft tissues (exception: commonly observed calcifications in soft tissues, such as the skin, kidney, tendons or vessels due to injury, disease, and aging, in the absence of systemic mineral imbalance).
• Current evidence of clinically significant corneal or retinal disorder. Comprehensive screening ophthalmologic examination is not necessary but may be conducted in order to evaluate subjects with a medical history of corneal or retinal disorders, based on investigator discretion. 
• Current use of prohibited medication as described in Section 5.6.2.
• Inability or unlikeliness to comply with the dose schedule and study evaluations, in the opinion of the investigator.
• Inadequate recovery from toxicity and/or complications from a major surgery before starting therapy.
• Inability to comprehend or unwilling to sign the ICF.