Concentration is an important property of independent random variable, showing that any reasonable function of such variables does not vary a lot around its mean. Observables generated by the iteration of a chaotic enough dynamical system often share a lot of properties with independent random variables. In this survey talk, we discuss several situations where one can prove concentration for them, in uniformly or non-uniformly hyperbolic situations. We also explain why such a property is important to answer relevant geometric or dynamical questions.
concentration - martingales - dynamical systems - Young towers - uniform hyperbolicity - moment bounds[-]

​Assume that a renormalized Birkhoff sum $S_n f/B_n$ converges in distribution to a nontrivial limit. What can one say about the sequence $B_n$? Most natural statements in the literature involve sequences $B_n$ of the form $B_n = n^\alpha L(n)$, where $L$ is slowly varying. We will discuss the possible growth rate of $B_n$ both in the probability preserving case and the conservative case. In particular, we will describe examples where $B_n$ grows superpolynomially, or where $B_{n+1}/B_n$ does not tend to $1$.[-]