Acyclic domains of linear orders: a survey

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.