We introduce a general framework that is based on distance semantics and investigate the main properties of the entailment relations that it induces. It is shown that such entailments are particularly useful for non-monotonic reasoning and for drawing rational conclusions from incomplete and inconsistent information. Some applications are considered in the context of belief revision, information integration systems, and consistent query answering for possibly inconsistent databases. Key words: paraconsistent and nonmonotonic reasoning, distance semantics, multiple-valued logics.