Due to the growing popularity of Description Logics-based knowledge representation systems, predominantly in the context of Semantic Web applications, there is a rising demand for tools offering non-standard reasoning services. One particularly interesting form of reasoning, both from the user as well as the ontology engineering perspective, is abduction. In this paper we introduce two novel reasoning calculi for solving ABox abduction problems in the Description Logic ALC, i.e. problems of finding minimal sets of ABox axioms, which when added to the knowledge base enforce entailment of a requested set of assertions. The algorithms are based on regular connection tableaux and resolution with set-of-support and are proven to be sound and complete. We elaborate on a number of technical issues involved and discuss some practical aspects of reasoning with the methods. © 2010 Springer Science+Business Media B.V.