Together with Newtonian mechanics, Maxwell electromagnetism, Einstein relativity and quantum mechanics, Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics, with uncountable applications in science and technology. This theory applies formidably well to a plethora of physical systems. Still, it fails in the realm of complex systems, characterized by generically strong space-time entanglement of their elements. On the basis of a nonadditive entropy (defined by an index $q$, which recovers, for $q=1$, the celebrated Boltzmann-Gibbs-von Neumann-Shannon entropy), it is possible to generalize the BG theory. We will briefly review the foundations and applications in natural, artificial and social systems.

A Bibliography is available at http://tsallis.cat.cbpf.br/biblio.htm