Semiflows are a class of Dynamical Systems, meaning that they help to describe how one state develops into another state over the course of time, a very useful concept in Mathematical Physics and Analytical Engineering. The authors concentrate on surveying existing research in continuous semi-dynamical systems, in which a smooth action of a real number on another object occurs from time zero, and the book proceeds from a grounding in ODEs through Attractors to Inertial Manifolds. The book demonstrates how the basic theory of dynamical systems can be naturally extended and applied to study the asymptotic behavior of solutions of differential evolution equations.