Given a dataset P and a preference function f, a top-k query retrieves the k tuples in P with the highest scores according to f. Even though the problem is well-studied in conventional databases, the existing methods are inapplicable to highly dynamic environments involving numerous longrunning queries. This paper studies continuous monitoring of top-k queries over a fixed-size window W of the most recent data. The window size can be expressed either in terms of the number of active tuples or time units. We propose a general methodology for top-k monitoring that restricts processing to the sub-domains of the workspace that influence the result of some query. To cope with high stream rates and provide fast answers in an on-line fashion, the data in W reside in main memory. The valid records are indexed by a grid structure, which also maintains book-keeping information. We present two processing techniques: the first one computes the new answer of a query whenever some of the current to...