Extreme value problems in random matrix theory and other disordered systems

We review some applications of central limit theorems and extreme values statistics in the context of disordered systems. We discuss several problems, in particular concerning random matrix theory and the generalization of the Tracy-Widom distribution when the disorder has `fat tails'. We underline the relevance of power-law tails for directed polymers and mean-field spin glasses and we point out various open problems and conjectures on these matters. We find that, in many instances, the assumption of Gaussian disorder cannot be taken for granted.