In this talk we will show the uniqueness of distributional solutions to 2D Navier-Stokes equations in vorticity form. As a consequence, one gets the uniqueness of probabilistically weak solutions to the corresponding McKean-Vlasov stochastic differential equations. It is also proved that for initial conditions with density in L4 these solutions are strong. In particular, one derives a stochastic representation of the vorticity of the fluid flow in terms of a solution to the McKean-Vlasov SDE. Finally, we show that the family of path laws of the solutions to McKean-Vlasov SDE form a nonlinear Markov process in the sense of McKean. This is in joint with Viorel Barbu and Michael Roeckner.