We address the problem of dealing with inconsistencies in Description Logic (DL) knowledge bases. Our general goal is both to study DL semantical frameworks which are inconsistency-tolerant, and to devise techniques for answering unions of conjunctive queries posed to DL knowledge bases under such inconsistency-tolerant semantics. Our work is inspired by the approaches to consistent query answering in databases, which are based on the idea of living with inconsistencies in the database, but trying to obtain only consistent information during query answering, by relying on the notion of database repair. We show that, if we use the notion of repair studied in databases, inconsistency-tolerant query answering is intractable, even for the simplest form of queries. Therefore, we study different variants of the repair-based semantics, with the goal of reaching a good compromise between expressive power of the semantics and computational complexity of inconsistency-tolerant query answering.