In the last two decades, multiobjective optimization has become mainstream because of its wide applicability in a variety of areas such engineering, management, the military and other fields. Multi-Objective Evolutionary Algorithms (MOEAs) play a dominant role in solving problems with multiple conflicting objective functions. They aim at finding a set of representative Pareto optimal solutions in a single run. Classical MOEAs are broadly in three main groups: the Pareto dominance based MOEAs, the Indicator based MOEAs and the decomposition based MOEAs. Those based on decomposition and indicator functions have shown high search abilities as compared to the Pareto dominance based ones. That is possibly due to their firm theoretical background. This paper presents state-of-the-art MOEAs that employ decomposition and indicator functions as fitness evaluation techniques along with other efficient techniques including those which use preference based information, local search optimizers, multiple ensemble search operators together with self-adaptive strategies, metaheuristics, mating restriction approaches, statistical sampling techniques, integration of Fuzzy dominance concepts and many other advanced techniques for dealing with diverse optimization and search problems