Keywords: Couette flow, Sommerfeld paradox, inviscid damping Abstract: Couette flows are shear flows with a linear velocity profile. First, They are known to be linearly stable for any Reynolds number, but become turbulent for large Reynolds numbers (Sommerfeld, 1908). With Charles Y. Li, we proposed an explanation of this paradox by constructing unstable shear flows arbitrarily close to Couette flows, for both inviscid and slightly viscous cases. Such unstable shears are possible seeds for the turbulent behaviors near Couette flows, as also supported by numerical work. Second, starting from the work of Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time. Such inviscid damping is open in the nonlinear level. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H1 norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H1 norm. Moreover, we showed that in (vorticity) H2 neighborhood of Couette flows, the only steady structures (including traveling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H2 space might be much simpler. We also obtained similar results for the problem of nonlinear Landau damping in 1D electrostatic plasmas.