  The cornerstone of Boltzmann-Gibbs ($BG$) statistical mechanics is the
Boltzmann-Gibbs-Jaynes-Shannon entropy $S_{BG} \equiv -k\int dx f(x)\ln f(x)$,
where $k$ is a positive constant and $f(x)$ a probability density function.
This theory has exibited, along more than one century, great success in the
treatment of systems where short spatio/temporal correlations dominate. There
are, however, anomalous natural and artificial systems that violate the basic
requirements for its applicability. Different physical entropies, other than
the standard one, appear to be necessary in order to satisfactorily deal with
such anomalies. One of such entropies is $S_q \equiv k (1-\int dx
[f(x)]^q)/(1-q)$ (with $S_1=S_{BG}$), where the entropic index $q$ is a real
parameter. It has been proposed as the basis for a generalization, referred to
as {\it nonextensive statistical mechanics}, of the $BG$ theory. $S_q$ shares
with $S_{BG}$ four remarkable properties, namely {\it concavity} ($\forall
q>0$), {\it Lesche-stability} ($\forall q>0$), {\it finiteness of the entropy
production per unit time} ($q \in \Re$), and {\it additivity} (for at least a
compact support of $q$ including $q=1$). The simultaneous validity of these
properties suggests that $S_q$ is appropriate for bridging, at a macroscopic
level, with classical thermodynamics itself. In the same natural way that
exponential probability functions arise in the standard context,power-law
tailed distributions, even with exponents {\it out} of the L\'evy range, arise
in the nonextensive framework. In this review, we intend to show that many
processes of interest in economy, for which fat-tailed probability functions
are empirically observed, can be described in terms of the statistical
mechanisms that underly the nonextensive theory.
