In this talk I will describe the progress that has been made so far
concerning the existence of global strong solutions to the
L^{2}-critical defocusing semilinear Schr\"odinger equation. A long
standing conjecture in the area is the existence
of a unique global strong L^{2}
solution to the equation that in addition scatters to a free solution as
time goes to infinity. I will demonstrate the proofs of partial results
towards an attempt for a final resolution of this conjecture.
I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces.
Many authors have contributed to the theory of this equation. I will convey my personal involment to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavlovic, G.
Staffilani, J. Colliander and M. Grillakis.