While most models of auctions and competitive bidding assume that each bidder's utility for an outcome depends only on his own profit, we allow the utility to also depend on any regret that a bidder suffers after the fact and characterize when and how this affects bidding. Specifically, a winner might regret "money left on the table" in Federal offshore oil lease sales, while a loser might regret seeing an object sell for less than his own value for it. We show that for risk-neutral bidders who, after the fact, know the winner's price for the object, a bidder's optimal bidding strategy will not depend on the relative weight given to profit versus regret if both forms of regret weigh equally heavily. However, if bidders weight money-left-on-the-table regret more heavily, then the bid taker suffers. Likewise, if losers do not learn the winner's price, then the bid taker also suffers. Thus, the existing models' exclusion of regret from risk neutral bidders' utility functions affects the applicability of the resulting theory under certain, now clearly delineated, conditions.