Starting with the work of Lebowitz-Rose-Speer in 1988, there has been great interest and significant progress in establishing invariance of certain Gibbs-type measures with respect to dispersive equations in 1 and 2 dimensions.  The invariant measures in question are absolutely continuous with respect to the free field,  and thus concentrated on functions of low regularity. N. Tzvetkov recently inituated the study of the absolute continuity of the flow at positive times with respect to measures supported on Sobolev spaces of arbitrary regularity, of type "Z^-1 exp (- ||u||_s^2)". This delicate property is known as quasi-invariance.  We will present a result establishing quasi-invariance of the 4th order cubic nonlinear Schroedinger equation, and explain why the situation is radically different in the absence dispersion.  (Joint work with N. Tzvetkov and T. Oh.)