This paper explores an HCI approach to designing markets, with a primary focus on peer-to-peer exchange platforms. We draw on recent work in economics that has documented how markets function, how they can be evaluated, and what can be done to fix them when they fail. We introduce five key concepts from market design: thickness, congestion, stability, safety, and repugnance. These lend HCI an analytic vocabulary for understanding why markets may succeed or struggle. Building on prior empirical work, we apply these concepts to compare two well-known network hospitality platforms, Couchsurfing and Airbnb. As a second illustrative case, we use market design to shed light on the challenges experienced by smaller-scale peer- to-peer marketplaces for lending, renting, and selling physical goods. To conclude, we discuss how this kind of analysis can make conceptual, evaluative, and generative contributions to the study and design of exchange platforms and other socio-technical systems.