Stability of coherent structures in two dimensional Euler equations, such as shear flows and vortices, is an important problem and a classical topic in fluid dynamics. Full nonlinear asymptotic stability results are difficult to obtain since the rate of stabilization is slow, the convergence of vorticity occurs only in weak, distributional sense, and the nonlinearity is strong. In a breakthrough work, Bedrossian and Masmoudi proved the first nonlinear asymptotic stability result, near the Couette flow (linear shear). In this talk, we will explain the physical relevance of the problem, survey recent progresses and in particular discuss our results proving the nonlinear asymptotic stability of general monotonic shear flows. If time permits, further open problems in the area will also be mentioned. This is joint work with Alexandru Ionescu. Speaker(s): Hao Jia (Univ. of Minnesota)