Abstract: Many analysis and monitoring applications require the repeated execution of expensive functions over streams of rapidly changing data. These applications appear in fields as varied as finance, supply chain management, and power utility monitoring. While many of these applications can be expressed declaratively as continuous queries (CQs) over streaming data, current continuous query processors are not designed to optimize queries with expensive user-defined functions. In my thesis, I propose that a large obstacle to efficient function execution is the treatment of functions as 'black boxes', where the query processor has no control over the processing inside each invocation. To deal with this problem, my thesis work provides new query processing technology which exploits new function interfaces to change how function calls execute in order to enhance performance. This thesis presents multiple query processing techniques that correspond to the new function APIs. In the first system presented, the CASPER caching system computes and caches ranges of data values for which the system knows the results of expensive predicates. Second, the Variable Accuracy Operators (VAOs) provide continuous query operators that use an interface which exposes the trade-off between work and accuracy inherent in many functions. To complete the thesis, the UFFISH system extends VAOs with a new caching technique inspired by CASPER.