Many web databases are hidden behind restrictive form-like interfaces which may or may not provide domain information for an attribute. When attribute domains are not available, domain discovery becomes a critical challenge facing the application of a broad range of existing techniques on third-party analytical and mash-up applications over hidden databases. In this paper, we consider the problem of domain discovery over a hidden database through its web interface. We prove that for any database schema, an achievability guarantee on domain discovery can be made based solely upon the interface design. We also develop novel techniques which provide effective guarantees on the comprehensiveness of domain discovery. We present theoretical analysis and extensive experiments to illustrate the effectiveness of our approach. Categories and Subject Descriptors H.2.7 [Database Administration]; H.3.5 [Online Information Services]: Web-based services General Terms Algorithms, Measurement, Perform...