In the last two decades, the fast and elitist non-dominated sorting genetic algorithm (NSGA-II) has attracted extensive research interests, and it is still one of the hottest research methods to deal with multi-objective optimization problems. Considering the importance and wide applications of NSGA-II method, we believe it is the right time to provide a comprehensive survey of the research work in this area, and also to discuss the potential in the future research. The purpose of this paper is to summarize and explore the literature on NSGA-II and another version called NSGA-III, a reference-point based many-objective NSGA-II approach. In this paper, we first introduce the concept of multi-objective optimization and the foundation of NSGA-II. Then we review the family of NSGA-II and their modifications, and classify their applications in engineering community. Finally, we present several interesting open research directions of NSGA-II for multi-objective optimization.