Hydrodynamic stability is of fundamental importance in the mechanics of fluids and is mainly concerned with the problem of transition to turbulence. This text is devoted to publication of original research papers, research-expository and survey articles with emphasis on unsolved problems and open questions in the mathematical modelling and computational aspects of hydrodynamic stability. Review chapters on the mathematical modelling and numerical simulation aspects of hydrodynamic stability, the physical background, and the limitations of the modelling and simulation procedures, due to particular mathematical or computational methods used, are included. This book should be appropriate for use in research and in research related courses on the subject. It includes chapters on channel flows, non-parallel shear flows, thin-film flows, strong viscous shear flows, Gortler Vortices, bifurcations in convection, wavy film flows and boundary layers.