Today many applications deal with large quantities of data that take the form of a stream -- an ordered sequence of data items. Streams arise in many forms, including text files, genetic sequences, sound tracks, geological strata from drill holes, videotape, histories of sensor values, and so on. There is a serious need for a new approach in stream data processing. Today's database systems are not able to manage this data. The approach we propose to investigate develops database systems from the established paradigm of stream processing, in which transducers (functional transformations) operate on streams. Under this approach databases are treated as streams, and queries or data analyzers are compositions of transducers that transform input data streams to output streams as needed. The objective of this research is to clarify the foundations of the resulting stream database paradigm. Important problems include adapting database concepts to the stream processing environment, characterizing useful stream pattern match queries, formalizing the optimization of stream database transducers, and finding the limits of the stream database approach. Beyond general uses in stream data processing, this research will have applications in DBMS/KBMS integration, database systems supporting data exploration, and a variety of special purpose database systems - including systems that manage genetic sequences, event histories, and on-line data.