Abstract: Despite recent widespread deployment of differential privacy, relatively little is known about what users think of differential privacy. In this work, we seek to explore users' privacy expectations related to differential privacy. Specifically, we investigate (1) whether users care about the protections afforded by differential privacy, and (2) whether they are therefore more willing to share their data with differential private systems. Further, we attempt to understand (3) users' privacy expectations of the differential private systems they may encounter in practice and (4) their willingness to share data in such systems. To answer these questions, we use a series of rigorously conducted surveys ($n=2424$). We find that users care about the kinds of information leaks against which differential privacy protects and are more willing to share their private information when the risks of these leaks are less likely to happen. Additionally, we find that the ways in which differential privacy is described in the wild haphazardly set users' privacy expectations, which can be misleading depending on the deployment. We synthesize our results into a framework for understanding a user's willingness to share information with differentially private systems, which takes into account the interaction between the user's prior privacy concerns and how differential privacy is described to them.