Graph structure is a flexible concept covering many differenttypes of graph properties. Hierarchical decompositions yielding thenotions of tree-width and clique-width, expressed by terms written withappropriate graph operations and associated with Monadic Second-orderLogic are important tools for the construction of Fixed-ParameterTractable algorithms and also for the extension of methods and resultsof Formal Language Theory to the description of sets of finite graphs.This informal overview presents the main definitions, results and openproblems and tries to answer some frequently asked questions.