This course examines the theory and physical mechanisms of hydrodynamic instability, with emphasis on the onset and evolution of disturbances in fluid flows and their role in transition to turbulence. Topics include linear stability theory, normal-mode analysis, energy methods, absolute and convective instabilities, nonlinear perturbation methods, and bifurcation theory. These concepts are applied to canonical flow instabilities, including Kelvin Helmholtz, Rayleigh Taylor, Rayleigh Bnard, and centrifugal instabilities. Students will develop the analytical tools needed to assess flow stability and interpret the mechanisms governing disturbance growth and nonlinear evolution.