The goal of this project is to study a few questions related to random perturbations of dynamical systems. These problems are at the center of the growing field of statistical properties of dynamical systems with a particular emphasis on the stability of the measures describing them. The big effort is nowadays oriented towards the investigation of systems with 'weak' mixing properties: non-uniformly and partially hyperbolic systems, isometries and billards. Our proposal fits with this orientation. The first goal of our project is to study the random perturbations of a class of non-uniformly expanding maps in any dimension.The second pourpose will be to study a relatively new field, that of piecewise random isometries with possible extension to some arithmetic properties of the interval exchange transformations. Thirdly we would like to investigate some random perturbations of billiards where the scatterers are permanently randomly deformed: the question here is to establish statistical properties such as ergodicity, decay of correlations and limit theorems. We will then be interested in random iterated function systems in the presence of infinitely many maps. Finally we propose to extend to the random setting a few results for the recurrence in deterministic dynamical systems. Our groups are in contact and collaborate with major researches in the field, and the aim of this proposal is to improve these interactions and contribute to the development of the field. Our partners are in England, Brazil, United States, Hungary, Italy, Portugal, Sweden and in several laboratories in France.