This paper reviews equilibrium behavior in different auction mechanisms. We will deal with two types of open auctions, English and Dutch, and two types of sealed-bid auctions, first-price and second-price, when there is a single object for sale and bidders have private values. We show that under certain conditions all four auctions yield the same expected revenue to the seller, but once these assumptions are relaxed revenue equivalence does not hold. We will also study auctions by using standard tools from demand theory. Finally, we will analyze collusive behavior of bidders. The two goals that an auction mechanism has to achieve are efficient allocation and maximization of the seller's expected revenue.