  In this paper, we review several recent results concerning well-posedness of
the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real
line R and on the circle T for solutions below the L^2-threshold. We point out
common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T,
suggesting that WNLS may be an appropriate model for the study of solutions
below L^2(T). In particular, in contrast with a recent result of Molinet who
proved that the solution map for the periodic cubic NLS equation is not weakly
continuous from L^2(T) to the space of distributions, we show that this is not
the case for WNLS.
