Hydrodynamic Instability Analysis provides a working knowledge of the stability of flows satisfying Navier-Stokes equations, the flow domain, boundary conditions and external forces are fixed. Solutions to equations vary with the viscosity and the initial conditions. The book is dedicated to important hydrodynamic instabilities types: Rayleigh-Taylor and Kelvin-Helmholtz. Different types of liquid jets instabilities are also considered. In the introductory chapter, fundamental concepts and properties of fluids are discussed as well as fundamental equations of mass, momentum and energy, and stability concepts. Emphasizing the jets instability rather than other aspects of fluid motion, in other chapters and appendices authors describe various important behaviors of fluids such as fundamental equations of the motion, deducing of Navier-Stokes equation, vortex motions, perturbation theory elements, special functions and their applications. In addition, this book incorporates an appendix with the Maple programs. The book will be useful to scientists, mathematicians and physicists, engineers and students, specializing in fluid mechanics and its applications.