•We analyze the origin of feedback mechanism with minimum adjustment or cost (FMMA/C).•We review FMMA/C in classical group decision making problems.•We review FMMA/C in complex group decision making problems.•We propose some open problems on FMMA/C. Consensus reaching process is a very powerful decision tool to eliminate the preference conflict in group decision making. In general, the consensus is achieved by the decision makers modifying their preferences (or opinions) toward a point of mutual consent, and the feedback mechanism aims to provide preference-modifications suggestions. In many situations, the preference-modifications mean cost and the resources for the consensus reaching process are limited. So, in the last decade, the feedback mechanism with minimum adjustment or cost (FMMA/C) has been developed and widely used in various group decision making contexts to improve consensus efficiency. In this review, we first analyze the origin and basic research paradigm of the FMMA/C. Then, we review the FMMA/C in two decision contexts: (1) the FMMA/C in classical group decision making problems, and (2) the FMMA/C in complex group decision making problems (e.g., social network, large-scale, and opinion dynamic group decision making problems). Finally, we identify research challenges and propose future research direction.