Local theories of finite groups

p-local theory of finite groups (which amounts to studying finite groups from the point of view of a chosen prime number p) is a set of tools and results which have been essentially developed during the period of heavy work towards the classification of finite simple groups. We shall give an overview on the subject, opening to the more contemporary extensions in terms of modular representations (representations in characteristic p). We shall see how results which may seem peculiar when viewed in the old terms give rise to general results or conjectures when put in the language of representations and derived categories.