The Poincaré recurrence theorem played a crucial role in the development of statistical mechanics at the end of the 19th century. In more recent years, Poincaré recurrences received a growing attention as a subject of the theory of dynamical systems.
In this respect, I will present a study of the statistics of first return times for a skew integrable map and for mixed dynamical systems. I will consider, in particular, a model obtained by coupling a regular and a mixing map together, showing that it allows to understand the behaviour of Poincaré recurrences for a system of more physical interest, the so called \"standard map\".
I will also describe an extension of the notion of statistics of first return times, namely the distributions of the number of visits, whose definition is based on successive return times. Finally, I will present an application of Poincaré recurrences to the analysis of the coding and noncoding regions of genomic sequences.