Abstract: The two dimensional incompressible Euler equation is an important model in the study of fluid dynamics for perfect fluid and viscous fluid flow in high Reynolds number regime. Despite its simplicity in formulation, the longtime behavior of solutions are very difficult to understand, due to the lack of global relaxation mechanism. In this talk, we will review some recent significant results in the study of precise dynamics in perturbative regimes near shear flows and vortices, which establish nonlinear asymptotic stability and also “weak but not strong” convergence to steady states, a manifestation of 2d turbulence. If time permits, we will also outline a new result towards nonlinear vortex symmetrization near general vortices. This is based on joint work with Alexandru Ionescu.