Matching dependencies (MDs) have been recently introduced as declarative rules for entity resolution (ER), i.e. for identifying and resolving duplicates in relational instance $D$. A set of MDs can be used as the basis for a possibly non-deterministic mechanism that computes a duplicate-free instance from $D$. The possible results of this process are the clean, "minimally resolved instances" (MRIs). There might be several MRIs for $D$, and the "resolved answers" to a query are those that are shared by all the MRIs. We investigate the problem of computing resolved answers. We look at various sets of MDs, developing syntactic criteria for determining (in)tractability of the resolved answer problem, including a dichotomy result. For some tractable classes of MDs and conjunctive queries, we present a query rewriting methodology that can be used to retrieve the resolved answers. We also investigate connections with "consistent query answering", deriving further tractability results for MD-based ER.