In an auction with externalities, a buyer's type is multidimensional and specifies the payoff he would get for each of theN+1 possible outcomes: the seller keeps the object or buyeri(i=1, …, N) gets the object. We provide a characterization of multidimensional incentive compatible mechanisms similar to that for one-dimensional mechanisms. Although reservation utilities are endogenous and type-dependent, the participation constraint is binding for only one “critical” type. A main difficulty in a multidimensional setting is the “integrability” condition. We present a geometric characterization for discontinuous conservative vector fields. In auctions where the buyers submit scalar bids and the seller transfers the object to one of the buyers for sure, a second-price auction maximizes revenue. With two buyers, this auction remains optimal even if the seller can set a reservation price.