One of the main applications of description logics is the ontology-based data access model, which requires algorithms for query answering over ontologies. In fact, some description logics, like those in the DL-Lite family, are designed so that simple queries, such as conjunctive queries, are efficiently computable. In this paper we study counting aggregate queries over ontologies, i.e. queries which use aggregate functions COUNT and COUNT DISTINCT. We propose an intuitive semantics for certain answers for these queries, which conforms to the open world assumption. We compare our semantics with other approaches that have been proposed in different contexts. We establish data and combined computational complexity for the problems of answering counting aggregate queries over ontologies for several variants of DL-Lite.