We study the power of selling opaque products, that is, products where a feature (such as color) is hidden from the customer until after purchase. Opaque products, which are sold with a price discount, have emerged as a powerful vehicle to increase revenue for many online retailers and service providers that offer horizontally differentiated items. In the opaque selling models we consider, all of the items are sold at a single common price alongside opaque products that may correspond to various subsets of the items. We consider two types of customers, risk-neutral ones, who assume they will receive a truly random item of the opaque product, and pessimistic ones, who assume they will receive their least favorite item of the opaque product. We benchmark opaque selling against two common selling strategies: discriminatory pricing, where one explicitly charges different prices for each item, and single pricing, where a single price is charged for all the items. We give a sharp characterization of when opaque selling outperforms discriminatory pricing; namely, this result holds for situations where all customers are pessimistic or the item valuations are supported on two points. In the latter case, we also show that opaque selling with just one opaque product guarantees at least 71.9% of the revenue from discriminatory pricing. We then provide upper bounds on the potential revenue increase from opaque selling strategies over single pricing and describe cases where the increase can be significantly more than that of discriminatory pricing. Finally, we provide pricing algorithms and conduct an extensive numerical study to assess the power of opaque selling for a variety valuation distributions and model extensions. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.