We address the problem of computing approximate answers to continuous sliding-window joins over data streams when the available memory may be insufficient to keep the entire join state. One approximation scenario is to provide a maximum subset of the result, with the objective of losing as few result tuples as possible. An alternative scenario is to provide a random sample of the join result, e.g., if the output of the join is being aggregated. We show formally that neither approximation can be addressed effectively for a slidingwindow join of arbitrary input streams. Previous work has addressed only the maximum-subset problem, and has implicitly used a frequencybased model of stream arrival. We address the sampling problem for this model. More importantly, we point out a broad class of applications for which an age-based model of stream arrival is more appropriate, and we address both approximation scenarios under this new model. Finally, for the case of multiple joins being execute...