This paper, forthcoming in Journal of Economic Surveys, provides an elementary, non-technical, survey of auction theory, by introducing and describing some of the critical papers in the subject. (The most important of these are reproduced in a companion book, The Economic Theory of Auctions, Paul Klemperer (ed.), Edward Elgar (pub.), 1999.) We begin with the most fundamental concepts, and then introduce the basic analysis of optimal auctions, the revenue equivalence theorem, and marginal revenues. Subsequent sections address risk-aversion, correlation and affiliation, asymmetries, entry costs and the number of bidders, collusion, multi-unit auctions, double auctions, royalties, incentive contracts, payments for quality, budget constraints, externalities, testing the theory, and other topics. Appendices contain technical details, some simple numerical examples, and a bibliography for each section.