Key Inclusion Criteria:
* Males or females ≥18 years of age who have provided written informed consent prior to completing any study-specific procedure. For Arm 2d, participants ≥12 years are eligible with informed consent/assent.
* Dose Escalation (Arm 1a): Participant has histologically confirmed locally advanced or metastatic cancer (cutaneous melanoma, NSCLC, non-muscle invasive bladder cancer, head and neck squamous cell carcinoma, microsatellite stable colorectal cancer \[MSS CRC\], basal cell carcinoma, or triple negative breast cancer) with measurable disease as determined by RECIST v1.1. Arm 1a participants must have received, and then progressed, relapsed, or been intolerant to, or ineligible for, at least 1 standard treatment regimen in the advanced or metastatic setting. Participants with a known driver mutation must have also received or been offered a mutation-directed therapy, where indicated. Participants must have a tumor lesion amenable to biopsy and must have another lesion that can be followed for response.
* Dose Confirmation (Arm 1b): Participant has histologically confirmed locally advanced or metastatic, and CPI refractory melanoma or locally advanced or metastatic, and CPI refractory NSCLC with measurable disease as determined by RECIST v1.1 who has disease progression after, at least 1 line of standard therapy (no limit to prior lines of therapy), and has been treated with or refused standard of care treatment. Participants in PD arm Group 2 must also have PD-L1 TPS ≥1%. Participants must have primary refractory or acquired secondary resistance to prior immune checkpoint treatments. Primary refractory is defined as prior exposure to anti-programmed death-1 (PD-1)/PD-L1 antibody for at least 6 weeks but no more than 6 months with demonstration of progression on 2 separate scans at least 4 weeks apart but no more than 12 weeks apart and progression occurring within 6 months after first dose of anti-PD-1 antibody. Acquired secondary resistance must have confirmed objective response or prolonged stable disease (SD) (\>6 months), followed by disease progression in the setting of ongoing treatment and confirmed progression on scans at least 4 weeks apart. Participants must have a tumor lesion amenable to biopsy and must have another lesion that can be followed for response.
1. For NSCLC participants with known EGFR, ALK, proto-oncogene tyrosine-protein kinase reactive oxygen species (ROS1), or other actionable mutations for which there are approved targeted therapies, participants must have received prior approved targeted therapy or have been offered and declined approved targeted therapy.
2. Expansion of the melanoma Arm 1b cohort (up to approximately 16 additional evaluable participants) will require prospective central testing of Screening tumor biopsies to confirm PD-L1 TPS ≥1% for eligibility.
* Dose Expansion Arms (Arm 2): Participant has histologically confirmed:
1. Arm 2a: Locally advanced or metastatic melanoma who have not yet received any prior systemic therapy for their melanoma in this setting.
2. Arm 2b: Newly diagnosed locally advanced or metastatic NSCLC with a PD-L1 TPS of ≥50% with no known EGFR or ALK positive tumor mutations who have not yet received any prior systemic therapy for their NSCLC (that is, treatment-naive).
3. Arm 2c: Locally advanced or metastatic melanoma in participants who have not received prior systemic therapy for melanoma in the advanced/metastatic setting.
4. Arm 2d: Advanced/metastatic melanoma that is CPI refractory and having a centrally confirmed PD-L1 TPS of ≥1% on their Screening tumor biopsy.
* All participants must have measurable disease as determined by RECIST v1.1.
* Participants must have a tumor lesion amenable to biopsy and must provide tumor biopsy sample at baseline (archival formalin-fixed, paraffin-embedded \[FFPE\] tissue collected within 90 days of informed consent is accepted as long as no intervening therapy is received, during this time), and optionally at all on-treatment timepoints (including response or progression) if medically feasible. Participants in Arm 2c: Sufficient tumor tissue (slides or FFPE block) for PD-L1 testing is required as per Laboratory Manual. Participants in Arm 2c may be replaced in this cohort if not PD-L1 evaluable. Participants in Arm 1b melanoma expansion cohort and Arm 2d: Central confirmation of PD-L1 TPS score is required prior to enrollment.
* If the participant is undergoing a new biopsy, they must have another lesion outside of the lesion biopsied at baseline that can be followed as a RECIST v1.1 target lesion for response.
* Participant has an Eastern Cooperative Oncology Group (ECOG) performance status of ≤1.
* Participant has adequate hematological and biological function.
Key Exclusion Criteria:
* Participant has active central nervous system tumors or metastases.
* Participant has received treatment with prohibited medications (that is, concurrent anticancer therapy including other chemotherapy, radiation \[local radiation for palliative care is permitted with approval from the Sponsor\], hormonal anticancer treatment, biologic therapy, or immunotherapy) or investigational agents within 5 half-lives or 14 days prior to the first day of study treatment, whichever is shorter. Prior exposure to any investigational or approved agent designed to simultaneously target both indoleamine 2,3-dioxygenase (IDO1) and PD-L1 is also exclusionary.
* Participant has required the use of additional immunosuppression (for example, infliximab) other than corticosteroids for the management of an AE, has experienced recurrence of an AE if rechallenged, and currently requires maintenance doses of \>10 milligrams (mg) prednisone or equivalent per day.
* Participant has any plan to receive a live attenuated vaccine during study treatment or has received a live vaccine within 30 days before the first dose of study treatment. Examples of live vaccines include, but are not limited to measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guérin, and typhoid vaccine. Seasonal influenza vaccines and non-live coronavirus disease 2019 (COVID-19) for injection are generally allowed.
* Participant has reversible toxicities from prior cancer therapy that have not recovered to Grade 1 or baseline. Any unresolved toxicity National Cancer Institute (NCI) Common Terminology Criteria for AEs (CTCAE) Grade ≥2 from previous anticancer therapy with the exception of alopecia, vitiligo, and prespecified laboratory values.
* Participant who is pregnant, breastfeeding, or is of childbearing potential, defined as those who are capable of becoming pregnant who are not willing to employ a highly effective method of contraception during dosing and for 90 days after the last dose of mRNA-4359 or 4 months after the last dose of pembrolizumab or 5 months after the last dose of ipilimumab/nivolumab administrations, whichever is longer.
* Sexually active participants who refuse to use a condom during intercourse or participants who will not refrain from sperm donation while taking study treatment and for 90 days after the last dose of mRNA-4359 or 4 months after the last dose of pembrolizumab or 5 months after the last dose ipilimumab/nivolumab, whichever is longer, or who will not refrain from sperm donation for the same time period.
* Participant has any unstable or clinically significant concurrent medical condition (for example, substance abuse, uncontrolled intercurrent illness including active infection, arterial thrombosis, and symptomatic pulmonary embolism) that would, in the opinion of the Investigator, jeopardize the safety of a participant, impact their expected survival through the end of the study participation, and/or impact their ability to comply with the protocol. Also including but not limited to, ongoing or active infection, interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea, active gastrointestinal bleeding or hemoptysis or history of bleeding disorder, or psychiatric illness/social situations that would limit compliance with study requirement, substantially increase risk of incurring AEs, or compromise the ability of the participant to give written informed consent.
* Participant has concurrent enrollment in another clinical study (unless it is an observational noninterventional clinical study).
