Over the past decades there has been a plausible interest for the generalized entropies that go beyond Boltzmann-Gibbs one. Thus, the information theory community has been primarily devoted to the analysis of their usefulness in the contexts of source and channel coding, as well as on the axiomatic characterization which approves their usefulness as the measures of information. On the other hand, the statistical physics community focused its efforts into developing the generalized thermostatistics and its applications in non-Boltzmann-Gibbs distributed systems and in the systems which exhibit non-extensive behavior.
 
In this talk we will review some of the existing definitions of generalized entropies, taking into account their information-theoretic and thermostatistical properties. We will consider their applications in communication theory, combinatorial optimization problems and complex systems and neural networks modeling. Recent results will be reviewed and open questions and possible research directions will be presented.