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Peer-to-peer markets such as eBay, Uber, and Airbnb allow small suppliers to compete with traditional providers of goods or services. We view the primary function of these markets as making it easy for buyers to find sellers and engage in convenient, trustworthy transactions. We discuss elements of market design that make this possible, including search and matching algorithms, pricing, and reputation systems. We then develop a simple model of how these markets enable entry by small or flexible suppliers, and the resulting impact on existing firms. Finally, we consider the regulation of peer-to-peer markets, and the economic arguments for different approaches to licensing and certification, data and employment regulation.