The promise of modern online marketplaces is that they lower the frictions for would-be trading partners to find and transact with each other. Such marketplaces enjoy, in principle, a virtuous feedback loop: by observing the outcomes of previous matches among participants, they learn how to enable better search and matching in the future. Modern online marketplaces rely on reputation systems to assess participants based on previous transactions. In this talk, we survey the design of these systems in practice, and go over some key challenges that they face. We then present some recent work towards a principled approach to understanding the design of these systems. Joint work with Nikhil Garg (Stanford).