  In this paper, we investigate both deterministic and stochastic 2D Navier
Stokes equations with anisotropic viscosity. For the deterministic case, we
prove the global well-posedness of the system with initial data in the
anisotropic Sobolev space $\tilde{H}^{0,1}.$ For the stochastic case, we obtain
existence of the martingale solutions and pathwise uniqueness of the solutions,
which imply existence of the probabilistically strong solution to this system
by the Yamada-Watanabe Theorem.
