Nonextensive statistics: Theoretical, experimental and computational evidences and connections

The domain of validity of standard thermodynamics and Boltzmann-Gibbs stati stical mechanics is discussed and then formally enlarged in order to hopefu lly cover a, variety of anomalous systems. The generalization concerns none xtensive systems, where nonextensivity it; understood in the thermodynamica l sense. This generalization was first proposed in 1988 inspired by the pro babilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computati onal evidences and connections, as well as some perspectives for the future . In addition to these, here and there we point out various (possibly) rele vant questions, whose answer would certainly clarify our current understand ing of the foundations of statistical mechanics and its thermodynamical imp lications.