On the derivation of the Gross-Pitaevskii equation

This article reflects in its content the talk the author gave at the XVII Congresso dell'Unione Matematica Italiana, held in Milano, 8-13 September 2003. We review about some recent results on the problem of deriving the Gross-Pitaevskii equation in dimension one from the dynamics of a quantum system with a large number of identical bosons. We explain the motivations for some peculiar choices (shape of the interaction potential, scaling, initial datum). Open problems are pointed out and difficulties and hindrances in replicating the strategy in higher dimension are put in evidence.