We analyze optimal auction design in the presence of negative externalities. We assume that externalities are a function of both the valuation of the agent who suffers it and the valuation of the agent who obtains the good. This introduces two technical difficulties: the reservation utility of each bidder becomes type-dependent and the equilibrium utility is not necessarily increasing in the agent's valuation. We characterize the properties of the optimal mechanism when externalities are

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... ly decreasing", "weakly decreasing" and "increasing" in the agent's valuation. Last, we discuss its implementation with sealed-bid auctions. . I am grateful to Juan Carrillo and very specially to an anonymous referee for useful comments.