Auction theory, as a subfield of economics, provides useful tools to model, analyze, and optimize radio resource management in cognitive radio environments. By using an auction, radio resources such as subchannel, time slot, and transmit power can be allocated among licensed and unlicensed users in the system, following market laws. Due to the flexibility of mechanism design, there are various auction mechanisms that have been applied to cognitive radio systems with different characteristics. In this article, we first provide an overview of the basics of general auctions. Then the motivations and specific design issues in applying auctions to wireless network architectures and protocols are discussed. Then we review the state of the art in the use of auction theory and mechanism design in cognitive radio networks. This will enable the readers to have a general view of auction fundamentals, as well as the recent development and applications of auction theory in the emerging cognitive wireless networks.