  This is a review on entropy in various fields of mathematics and science. Its
scope is to convey a unified vision of the classical as well as some newer
entropy notions to a broad audience with an intermediate background in
dynamical systems and ergodic theory. Due to the breadth and depth of the
subject, we have opted for a compact exposition whose contents are a compromise
between conceptual import and instrumental relevance. The intended technical
level and the space limitation born furthermore upon the final selection of the
topics, which cover the three items named in the title. Specifically, the first
part is devoted to the avatars of entropy in the traditional contexts: many
particle physics, information theory, and dynamical systems. This chronological
order helps to present the materials in a didactic manner. The axiomatic
approach will be also considered at this stage to show that, quite remarkably,
the essence of entropy can be encapsulated in a few basic properties. Inspired
by the classical entropies, further akin quantities have been proposed in the
course of time, mostly aimed at specific needs. A common denominator of those
addressed in the second part of this review is their major impact on research.
The final part shows that, along with its profound role in the theory, entropy
has interesting practical applications beyond information theory and
communications technology. For this sake we preferred examples from applied
mathematics, although there are certainly nice applications in, say, physics,
computer science and even social sciences. This review concludes with a
representative list of references.
