Abstract. What does a data stream mean? Much of the extensive work on query operators and query processing for data streams has proceeded without the benefit of an answer to this question. While such imprecision may be tolerable when dealing with simple cases, such as flat data, guaranteed physical order and element-wise operations, it can lead to ambiguities when dealing with nested data, disordered streams and windowed operators. We propose reconstitution functions to make the denotation and representation of data streams more precise, and use these functions to investigate the connection between monotonicity and nonblocking behavior of stream operators. We also touch on a reconstitution function for XML data. Other aspects of data stream semantics we consider are the use of punctuation to delineate finite subsets of a stream, adequacy of descriptions of stream disorder, and the formal specification of windowed operators. 1