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We propose an extension of the behavioral theory of interactive sequential algorithms to deal with the following situation. A query is issued during a certain step, but the step ends before any reply is received. Later, a reply arrives, and later yet the algorithm makes use of this reply. By a persistent query, we mean a query for which a late reply might be used. Our proposal involves issuing, along with a persistent query, a location where a late reply is to be stored. After presenting our proposal in general terms, we discuss the modifications that it requires in the existing axiomatics of interactive sequential algorithms and in the existing syntax and semantics of abstract state machines. To make that discussion self-contained, we include a summary of this material before the modifications. Fortunately, only rather minor modifications are needed.