In the past few years, methods such as those by J. Bourgain on the one hand and by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao on the other have been applied to study the global in time existence of dispersive equations at regularities which are right below or in between those corresponding to conserved quantities. However, for many dispersive equations and systems there still remains a gap between the local in time results and those that could be globally achieved. In those cases, it seem natural to return to one of Bourgain's early approaches where global in time existence was studied in the almost surely sense via the existence and invariance of the associated Gibbs measure. The failure to show the global existence often comes from certain `exceptional' initial data, and the virtue of the Gibbs measure is that it does not see that exceptional set. At the same time, the invariance of the Gibbs measure, just like the usual conserved quantities, controls the growth in time of those solutions in its support. The difficulty in this approach lies, of course, in the actual construction of the associated Gibbs measure and in showing its invariance under the flow. In this talk we will describe this approach in the context of the 1D periodic derivative nonlinear Schrodinger equation.