In recent years, following the seminal works of Villani and Mouhot on Landau damping, phase-mixing as a damping mechanism and inviscid damping in fluids have attracted much interest. In this talk, I will provide an introduction to the underlying mechanism and discuss new linear stability and damping results near Taylor-Couette flow between concentric cylinders. This is based on joint work with Michele Coti Zelati at Imperial College, London.