We study the problem of database repairs through a rule-based framework that we refer to as Delta Rules. Delta rules are highly expressive and allow specifying complex, cross-relations repair logic associated with Denial Constraints, Causal Rules, and allowing to capture Database Triggers of interest. We show that there are no one-size-fits-all semantics for repairs in this inclusive setting, and we consequently introduce multiple alternative semantics, presenting the case for using each of them. We then study the relationships between the semantics in terms of their output and the complexity of computation. Our results formally establish the tradeoff between the permissiveness of the semantics and its computational complexity. We demonstrate the usefulness of the framework in capturing multiple data repair scenarios for an academic search database and the TPC-H databases, showing how using different semantics affects the repair in terms of size and runtime, and examining the relationships between the repairs. We also compare our approach with SQL triggers and a state-of-the-art data repair system.