Abstract Establishing existence and characterizing equilibria are both important achievements in the study of auctions. However, we recognize that equilibria existence results form the basis for well accepted characterizations. In this survey, we review the landmark results and highlight open questions regarding equilibria existence and characterizations in auctions. In addition, we review the standard assumptions underlying these results, and discuss the suitability of the Nash equilibrium solution concept. We focus our review on single-object auctions, but also review results in multi-unit, divisible, combinatorial and double auctions.