Online applications that are open to participation lack reliable methods to establish the integrity of user-generated information. Users may unknowingly own compromised devices, or intentionally publish forged information. In these scenarios, applications need some way to determine the "correctness" of autonomously generated information. Towards that end, this thesis presents a "trust-but-verify" approach that enables open online applications to independently verify the information generated by each participant. In addition to enabling independent verification, our framework allows an application to verify less information from more trustworthy users and verify more information from less trustworthy ones. Thus, an application can trade-off performance for more integrity, or vice versa. We apply the trust-but-verify approach to three different classes of online applications and show how it can enable 1) high-integrity, privacy-preserving, crowd-sourced sensing 2) non-intrusive cheat detection in online games, and 3) effective spam prevention in online messaging applications.