This paper provides a recent overview of the exact, approximate, and hybrid optimization methods that handle Multi-Objective Mixed-Integer Non-Linear Programming (MO-MINLP) problems. Both the domains of exact and approximate research have experienced significant growth, driven by their shared goal of addressing a wide range of real-world problems. This work presents a comprehensive literature review that highlights the significant theoretical contributions in the field of hybrid approaches between these research areas. We also point out possible research gaps in the literature. Hence, the main research questions to be answered in this paper involve the following: (1) how to exactly or approximately solve a MO-MINLP problem? (2) What are the drawbacks of exact methods as well as approximate methods? (3) What are the research lines that are currently underway to enhance the performances of these methods? and (4) Where are the research gaps in this field? This work aims to provide enough descriptive information for newcomers in this area about the research that has been carried out and that is currently underway concerning exact, approximate, and hybrid methods used to solve MO-MINLP problems.