Fraudulently posted online rental listings, rental scams, have been frequently reported by users. However, our understanding of the structure of rental scams is limited. In this paper, we conduct the first systematic empirical study of online rental scams on Craigslist. This study is enabled by a suite of techniques that allowed us to identify scam campaigns and our automated system that is able to collect additional information by conversing with scammers. Our measurement study sheds new light on the broad range of strategies different scam campaigns employ and the infrastructure they depend on to profit. We find that many of these strategies, such as credit report scams, are structurally different from the traditional advanced fee fraud found in previous studies. In addition, we find that Craigslist remove less than half of the suspicious listings we detected. Finally, we find that many of the larger-scale campaigns we detected depend on credit card payments, suggesting that a payment level intervention might effectively demonetize them.