This book chapter is published open access.
The hydrodynamic stability theory is mainly concerned with how laminar flows become unstable and transit to turbulence at high Reynolds number. To shed some light on the transition mechanism, Trefethen et al. [Science 261(1993)] proposed the transition threshold problem: how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. Many effects such as 3D lift-up, inviscid damping, enhanced dissipation, and boundary layer play a crucial role in determining the transition threshold. In this note, we will first survey some important progress on linear inviscid damping and enhanced dissipation for shear flows. Then we will outline key ingredients in our proof of transition threshold for the 3D Couette flow in a finite channel.