My PhD was on so called "pure" model theory, and my advisor was not very much interested in applications of model theory to algebra. Now I feel the need to fill in the gap, and I'd like to educate myself on applied model theory. One of the questions I always asked myself is about the study of groups definable (or interpretable) in some structures : it seems that model theorists are very fond of that, since a long time ago (at least since the 70's to my knowledge, and even quite recently with works about groups definable in NIP structures by Pillay). Why is it so ? I guess that the origin of all this is the fact that groups definable in ACF are precisely algebraic groups. This is indeed a nice link between model theory and algebraic geometry (by the way, does this link has brought something interesting and new to algebraic geometry, or has it always been just a slightly different point of view ?). But if it is so, why going on to study extensively groups definable on such exotic kind of theories as NIP or simple for example ? Is it only because something can be said about those groups and that groups are prestigious objects within mathematics, or are there deeper reasons ?