A Study of Fianlimab, Cemiplimab, and Ipilimumab in People With Melanoma

Inclusion Criteria: * Age ≥ 18 years at the time of informed consent * Patient/legal authorized representative (LAR) must be able to provide informed consent. * Patient must have a histologically confirmed diagnosis of locally advanced unresectable stage III/IV or metastatic stage IV cutaneous or mucosal melanoma that has progressed on PD-1/PD-L1 therapy: o For Cohort A, the patient's melanoma must have progressed on prior PD-1 monotherapy * For Cohort B, the patient's melanoma must have progressed on prior combination PD-1 + LAG-3 blockade * Note: Intervening lines of targeted therapy, chemotherapy, bispecific (e.g. IMCgp100) and cell-based therapies are permitted between last ICI-based therapy and the start of study therapy * Note: For cohort A, peptide and mRNA vaccines may have been combined with PD-1 monotherapy as long as no other checkpoint inhibitors were concomitantly administered. For cohort B, peptide and mRNA vaccines may have been combined with combined PD-1 + LAG-3 blockade as long as no other checkpoint inhibitors were concomitantly administered Note: Prior PD-1 monotherapy (Cohort A) or PD-1 and LAG-3 blockade (Cohort B) may have been given in the neoadjuvant or adjuvant setting as long as progression is documented within 3 months of the final dose neoadjuvant/adjuvant therapy * Patients must have measurable disease as defined by RECIST v1.1 o Note: Lesions previously injected with Talimogene laherparepvec or other local therapies may not be selected as target lesions unless they have demonstrated subsequent growth after injection * If a suitable archival tissue sample is available, the patient must be willing to have this specimen submitted for research. If an archival sample is not available, the patient is still a candidate for the trial, and every reasonable effort will be made to obtain a biopsy if deemed safe * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 1 ° Adequate laboratory function at screening, defined as: ° Hemoglobin ≥ 10 gm/dL (≥ 6.2 mmol/L) ° Platelet count ≥ 100 × 10\^9 /L °Serum direct bilirubin ≤ 1.5 × ULN; AST and ALT ≤ 2.5 × ULN. (Total bilirubin \< 3 mg/dL for subjects with Gilbert's disease) * No signs of active coronary ischemia, including ECG changes or elevated troponin if clinically indicated * Calculated creatinine clearance (CrCl) ≥30 mL/min based on the Cockcroft-Gault equation * All immune-related adverse events (irAE's) from prior ICI based therapy must have improved to Grade 1 or lower with the following exception. Patients with stable adrenal insufficiency are eligible as long as 1) they are asymptomatic at the time of consent on a stable steroid replacement regimen and 2) are on daily steroid replacement of 7.5mg prednisone equivalent or less. * All women of childbearing potential (WOCBP)\* or sexually active men must practice highly effective contraception prior to the initial dose/start of the first treatment, during the study, and for at least 6 months after the last dose. Highly effective contraceptive measures in women include o Stable use of combined (estrogen and progestogen containing) hormonal contraception (oral, intravaginal, transdermal) or progestogen-only hormonal contraception (oral, injectable, implantable) associated with inhibition of ovulation initiated 2 or more menstrual cycles prior to screening, * Intrauterine device (IUD), * Intrauterine hormone-releasing system (IUS), * Bilateral tubal ligation, * Vasectomized partner,† and/or * Sexual abstinence.‡,§ * Male study participants with WOCBP partners are required to use condoms unless they are vasectomized† or practice sexual abstinence.‡,§ * \* WOCBP are defined as women who are fertile following menarche until becoming postmenopausal, unless permanently sterile. A postmenopausal state is defined as no menses for 12 months without an alternative medical cause. A high follicle stimulating hormone (FSH) level in the postmenopausal range may be used to confirm a postmenopausal state in women not using hormonal contraception or hormonal replacement therapy. However, in the absence of 12 months of amenorrhea, a single FSH measurement is insufficient to determine the occurrence of a postmenopausal state. The above definitions are according to Clinical Trial Facilitation Group (CTFG) guidance. Pregnancy testing and contraception are not required for women with documented hysterectomy or tubal ligation. Permanent sterilization methods include hysterectomy, bilateral salpingectomy, and bilateral oophorectomy. * Vasectomized partner or vasectomized study participant must have received medical assessment of the surgical success. * Sexual abstinence is considered a highly effective method only if defined as refraining from heterosexual intercourse during the entire period of risk associated with the study drugs. The reliability of sexual abstinence needs to be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. * Periodic abstinence (calendar, symptothermal, post-ovulation methods), withdrawal (coitus interruptus), spermicides only, and lactational amenorrhea method (LAM) are not acceptable methods of contraception. Female condom and male condom should not be used together. Exclusion Criteria: * Uveal melanoma * Untreated central nervous system (CNS) metastases or leptomeningeal involvement; patients with brain metastases definitively treated with surgery or stereotactic radiosurgery (SRS) are permitted * Receipt of the following prior therapies: * For Cohort A: Any prior anti-LAG-3 (e.g., relatlimab) or CTLA-4 (e.g., ipilimumab) directed therapy, unless it was given in the adjuvant or neoadjuvant setting and the last dose was given more than three months prior to disease recurrence * For Cohort B: Any prior CTLA-directed therapy (e.g., ipilimumab), unless it was given in the adjuvant or neoadjuvant setting and the last dose was given more than three months prior to disease recurrence * Prior Grade 3 or greater neurologic toxicity associated with a prior line of ICI therapy * Any prior myocarditis associated with ICI therapy * Prior Grade 3 or greater colitis or enteritis requiring hospitalization * Concurrent systemic steroid therapy higher than physiologic dose steroid replacement (\>7.5 mg/day of prednisone or equivalent), given within 14 days of starting treatment, or other immunosuppressive medications within 14 days of the start of treatment. Inhaled or topical steroids are permitted in the absence of active autoimmune disease. * Receipt of a live vaccine within 30 days of planned start of study medication * Significant infection requiring systemic antibiotics within 2 weeks of the planned start of study medication (e.g., pneumonia, cellulitis) * Uncontrolled (i.e., unstable) concomitant medical condition or organ system dysfunction which, in the treating Investigator's opinion, could compromise the patient's safety or compliance with the study procedures. * Other active, concurrent malignancy that requires ongoing systemic treatment or interferes with radiographic assessment of melanoma response as determined by the treating investigator * History of severe hypersensitivity reactions to any unknown allergens or any components of the study drugs (active ingredients or excipients) * Has uncontrolled infection with human immunodeficiency virus, hepatitis B, or hepatitis C infection; or has a diagnosis of immunodeficiency. Notes: * Patients will be tested for hepatitis C virus (HCV) and hepatitis B virus (HBV) at screening. * Patients with known HIV infection who have controlled infection (undetectable viral load (HIV RNA PCR) and CD4 count above 350 either spontaneously or on a stable antiviral regimen) are permitted. For patients with controlled HIV infection, monitoring will be performed per local standards. * Patients with hepatitis B (HBsAg+) who have controlled infection (serum hepatitis B virus DNA PCR that is below the limit of detection and receiving anti-viral therapy for hepatitis B) are permitted. Patients with controlled infections must undergo periodic monitoring of HBV DNA. Patients must remain on anti-viral therapy for at least 6 months beyond the last dose of investigational study drug. * Patients who are hepatitis C virus antibody positive (HCV Ab+) who have controlled infection (undetectable HCV RNA by PCR either spontaneously or in response to a successful prior course of anti-HCV therapy) may be enrolled into the study. * Patients who are breastfeeding or who are pregnant as evidenced by a positive serum pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) performed within 14 days of the first dose of study drug. * Prisoners or participants who are involuntarily incarcerated. (Note: Under certain specific circumstances where local regulations permit, a person who has been imprisoned may be permitted to continue as a participant.) * Participants who are compulsorily detained for treatment of either a psychiatric or physical illness (e.g., transmissible infection)