Explore the intricacies of hydrodynamic stability theory and the transition threshold problem in this 46-minute lecture by Zhifei Zhang for the International Mathematical Union. Delve into the mechanisms behind laminar flow instability and turbulence transition at high Reynolds numbers. Examine key physical effects such as 3-D lift-up, inviscid damping, enhanced dissipation, and boundary layer behavior. Survey recent advancements in linear inviscid damping and enhanced dissipation for shear flows. Gain insights into the proof of transition threshold for 3-D Couette flow in a finite channel, including key ideas and main components. Follow along with topics like Reynolds experiments, Navier-Stokes equations, eigenvalue analysis, subcritical transition, and various mathematical models. Conclude with an exploration of open problems in the field, providing a comprehensive overview of this complex area of fluid dynamics.