  We start by reviewing recent probabilistic results on ergodic sums in a large
class of (non-uniformly) hyperbolic dynamical systems. Namely, we describe the
central limit theorem, the almost-sure convergence to the gaussian and other
stable laws, and large deviations. Next, we describe a new branch in the study
of probabilistic properties of dynamical systems, namely concentration
inequalities. They allow to describe the fluctuations of very general
observables and to get bounds rather than limit laws. We end up with two
sections: one gathering various open problems, notably on random dynamical
systems, coupled map lattices and the so-called nonconventional ergodic
averages; and another one giving pointers to the literature about moderate
deviations, almost-sure invariance principle, etc.
