The notions of stability and instability play a very important role in mathematical physics and, in particular, in mathematical models of fluids flows. Currently, one of the most important problems in this area is to describe different kinds of instability, to understand their nature, and also to work out methods for recognizing whether a mathematical model is stable or instable. In the current volume, Claude Bardos and Andrei Fursikov, have drawn together an impressive array of international contributors to present important recent results and perspectives in this area. The main topics covered are devoted to mathematical aspects of the theory but some novel schemes used in applied mathematics are also presented.Various topics from control theory, free boundary problems, Navier-Stokes equations, first order linear and nonlinear equations, 3D incompressible Euler equations, large time behavior of solutions, etc. are concentrated around the main goal of these volumes the stability (instability) of mathematical models, the very important property playing the key role in the investigation of fluid flows from the mathematical, physical, and computational points of view. World - known specialists present their new results, advantages in this area, different methods and approaches to the study of the stability of models simulating different processes in fluid mechanics.