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Due to the lack of efficient native XML database management systems, XML data manipulation and query evaluation may be resource-consuming, and represent a bottleneck for several computationally intensive applications. To overcome the above limitations, a possible solution consists in computing synopsis data structures from XML databases, i.e. compressed representations providing a "succinct" description of the original databases while ensuring low computational overhead and high accuracy for many XML processing tasks. Specifically, these data structures are very useful for both selectivity estimation and approximate query answering purposes. On the other hand, while synopsis data structures have been widely applied to relational as well as multidimensional data, a full usage for XML data is still lacking. Inspired by these considerations, in this paper we discuss the models and issues of synopsis data structures for XML databases, and we complete our analysis by selecting and discussing future perspectives for this research field.