Abstract: Most optimization problems naturally have several objectives, usually in conflict with each other. The problems with two or three objective functions are referred to as Multi-Objective Problems (MOP). However, many real-world applications often involve four or more objectives, which are commonly recognized as many-objective optimization problems (MaOP). Multi and many-objective algorithms have a great application in engineering science. This study addresses a complete and updated review of the literature for multi and many-objective problems and discusses 32 more important algorithms in detail. Afterward, the ZDT and DLTZ benchmark problems for multi-objective test problems are reviewed. All methods have been studied under recent state-of-the-art quality measures. Moreover, we discuss the historical roots of multi-objective optimization, the motivation to use evolutionary algorithms, and the most popular techniques currently in use.