We consider infinite recursive (i.e., computable) relational data bases. Since the set of computable queries on such data bases is not closed under even simple relational operations, one must either make do with a very modest class of queries or considerably restrict the class of allowed data bases. We define two query languages, one for each of these possibilities, and prove their completeness. The first is the language of quantifier-free first-order logic, which is shown to be complete for the non-restricted case. The second is an appropriately modified version of Chandra and Harel's language QL, which is proved complete for the case of ``highly symmetric" data bases, i.e., ones whose set of automorphisms is of finite index for each tuple-width. We also address the related notion of BP-completeness.