Search auctions have become a dominant source of revenue generation on the Internet. Such auctions have typically used per-click bidding and pricing. We propose the use of hybrid auctions where an advertiser can make a per-impression as well as a per-click bid, and the auctioneer then chooses one of the two as the pricing mechanism. We assume that the advertiser and the auctioneer both have separate beliefs (called priors) on the click-probability of an advertisement. We first prove that the hybrid auction is truthful, assuming that the advertisers are risk-neutral. We then show that this auction is superior to the existing per-click auction in multiple ways: 1. We show that risk-seeking advertisers will choose only a per-impression bid whereas risk-averse advertisers will choose only a per-click bid, and argue that both kind of advertisers arise naturally. Hence, the ability to bid in a hybrid fashion is important to account for the risk characteristics of the advertisers. 2. For obscure keywords, the auctioneer is unlikely to have a very sharp prior on the click-probabilities. In such situations, we show that having the extra information from the advertisers in the form of a per-impression bid can result in significantly higher revenue. 3. An advertiser who believes that its click-probability is much higher than the auctioneer's estimate can use per-impression bids to correct the auctioneer's prior without incurring any extra cost. 4. The hybrid auction can allow the advertiser and auctioneer to implement complex dynamic programming strategies to deal with the uncertainty in the click-probability using the same basic auction. The per-click and per-impression bidding schemes can only be used to implement two extreme cases of these strategies. As Internet commerce matures, we need more sophisticated pricing models to exploit all the information held by each of the participants. We believe that hybrid auctions could be an important step in this direction. The hybrid auction easily extends to multiple slots, and is also applicable to scenarios where the hybrid bidding is per-impression and per-action (i.e. CPM and CPA), or per-click and per-action (i.e. CPC and CPA). Copyright is held by the International World Wide Web Conference Committee (IW3C2).