  Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{BG}=-k
\sum_{i=1}^W p_i \ln p_i$. It enables a successful thermal approach of
ubiquitous systems, such as those involving short-range interactions, markovian
processes, and, generally speaking, those systems whose dynamical occupancy of
phase space tends to be ergodic. For systems whose microscopic dynamics is more
complex, it is natural to expect that the dynamical occupancy of phase space
will have a less trivial structure, for example a (multi)fractal or
hierarchical geometry. The question naturally arises whether it is possible to
study such systems with concepts and methods similar to those of standard
statistical mechanics. The answer appears to be {\it yes} for ubiquitous
systems, but the concept of entropy needs to be adequately generalized. Some
classes of such systems can be satisfactorily approached with the entropy
$S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $q \in \cal R$, and $S_1
=S_{BG}$). This theory is sometimes referred in the literature as {\it
nonextensive statistical mechanics}. We provide here a brief introduction to
the formalism, its dynamical foundations, and some illustrative applications.
In addition to these, we illustrate with a few examples the concept of {\it
stability} (or {\it experimental robustness}) introduced by B. Lesche in 1982
and recently revisited by S. Abe.
