On the foundations of statistical mechanics

We briefly review the foundations and applications of statistical mechanics based on the nonadditive entropies S (q) . Then we address four frequently focused points, namely (i) On the form of the constraints within a variational entropy principle; (ii) Are the q-indices first-principle-computable quantities or fitting parameters?; (iii) If one admits violation of the entropic additivity, why not admitting also violation of the entropic extensivity?; and (iv) Critical-like behavior.