Key Inclusion Criteria:  Has a histologically- or cytologically-diagnosed solid tumor which is advanced or metastatic and which has progressed on or following at least one systemic therapy regimen administered for advanced or metastatic disease or for which no approved therapy exists. Subject's prior treatment should include all approved regimens that have demonstrated a survival advantage for the subject's disease, stage, and line of therapy. Is aged ≥18 years at the time of signing the ICF Has provided written informed consent Has an ECOG Performance Status of 0 or 1 Has measurable disease per RECIST 1.1 Has an adequate tumor sample. Has adequate liver, renal, hematologic, pulmonary, cardiac, and coagulation function. Has a negative serum pregnancy test (for women of child-bearing potential) at Screening and a negative urine pregnancy test on Day 1 prior to the first dose of MPT 0118 Ability to swallow and retain and absorb oral medications in tablet or crushed form orally or via feeding tube (e.g., nasogastric feeding tube or percutaneous endoscopic gastrostomy feeding tube)  Key Exclusion Criteria:  Has received cytotoxic chemotherapy, biologic agent, investigational agent, checkpoint inhibitors, or radiation therapy ≤3 weeks prior to the first dose of MPT-0118 Has received small-molecule kinase inhibitors or hormonal agents ≤14 days prior to the first dose of MPT-0118 Has been previously treated with a MALT1 inhibitor Has clinically significant AEs that have not returned to baseline or ≤Grade 1 based on National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) v5.0 Has received systemic immunosuppressive agents within 14 days of the first dose of MPT-0118 Has undergone major surgery ≤6 weeks or minor surgery ≤14 days prior to the first dose of MPT-0118 Has clinically significant intercurrent disease Part B and Part C: Has previously been treated with PD-1, PD-L1, or CTLA-4 inhibitors and required dose-interruption, permanent discontinuation, or systemic immunosuppression due to immune-related AEs Has primary central nervous system (CNS) tumors or brain or leptomeningeal metastasis. Has human immunodeficiency virus (HIV) infection Has active hepatitis B or C infection Women who are pregnant or breastfeeding Has an unwillingness or inability to comply with procedures required in this protocol Is currently receiving any other anticancer or investigational agent