In spite of their widely recognized importance in the framework of the modern understanding of the asteroidal population and its collisional evolution, asteroid dynamical families have long been a puzzling subject of research, due to the disagreement among the family lists published by different authors. In the present review, the definition and meaning of asteroid families are critically discussed, as well as the various problems which have to be faced by any suitable technique of family identification. In this respect, major improvements have been achieved during the last few years. The most recent family searches show an excellent agreement both in the number of reliably identified families, and in their members. Moreover, the overall performances of the most recent techniques of family identification have been tested by means of numerical simulations, with encouraging results. For these reasons, we believe that we are presently at the beginning of an era in which detailed physical studies of families can be attempted, and observational campaigns can be planned on the basis of solid evidence, like in the case of the recent spectacular results obtained for the family associated to the large asteroid 4 Vesta.