  In this expository note we discuss our recent work [arXiv:1306.5028] on the
nonlinear asymptotic stability of shear flows in the 2D Euler equations of
ideal, incompressible flow. In that work it is proved that perturbations to the
Couette flow which are small in a suitable regularity class converge strongly
in $L^2$ to a shear flow which is close to the Couette flow. Enstrophy is mixed
to small scales by an almost linear evolution and is generally lost in the weak
limit as t -> +/- infinity. In this note we discuss the most important physical
and mathematical aspects of the result and the key ideas of the proof.
