  Clausius introduced, in the 1860s, a thermodynamical quantity which he named
{\it entropy} $S$. This thermodynamically crucial quantity was proposed to be
{\it extensive}, i.e., in contemporary terms, $S(N) \propto N$ in the
thermodynamic limit $N \to\infty$. A decade later, Boltzmann proposed a
functional form for this quantity which connects $S$ with the occurrence
probabilities of the microscopic configurations (referred to as {\it
complexions} at that time) of the system. This functional is, if written in
modern words referring to a system with $W$ possible discrete states,
$S_{BG}=-k_B \sum_{i=1}^W p_i \ln p_i$. The BG entropy is {\it additive},
meaning that, if A and B are two probabilistically independent systems, then
$S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)$. The words, {\it extensive} and {\it
additive}, were practically treated, for over more than one century, as almost
synonyms, and $S_{BG}$ was considered to be the unique form that $S$ could
take. In other words, the functional $S_{BG}$ was considered to be universal.
It has become increasingly clear today that it is {\it not} so, and that those
two words are {\it not} synonyms, but happen to coincide whenever we are
dealing with paradigmatic Hamiltonians involving {\it short-range} interactions
between their elements, presenting no strong frustration and other
"pathologies". These facts constitute the basis of a generalization of the BG
entropy and statistical mechanics, introduced in 1988, and frequently referred
to as nonadditive entropy $S_q$ and nonextensive statistical mechanics,
respectively. We briefly review herein these points, and exhibit recent as well
as typical applications of these concepts in natural, artificial, and social
systems, as shown through theoretical, experimental, observational and
computational predictions and verifications.
