The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cove r a v ariety o f anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was rst proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present e ort, after introducing some historical background, we brie y describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various possibly relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.