Execution of XML query languages represents a critical service of any XML data management system. Though the backgrounds of different approaches range from relational, through object-oriented to native data architectures, all of them need specialized operations and/or storage structures to complete the retrieval tasks efficiently. In this article, we survey the most important approaches, which are mainly based on specialized implementations of the containment joins and edge mapping. We also present a new approach to query evaluation. It is based on miniaturized XML tree representations, called signatures, consisting of node names and their tree structural relationships. We demonstrate how such signatures can be used for efficient tree navigation and query twig pattern matching in large XML databases. Other applications are briefly discussed in the conclusions.