In this book, dynamical systems are investigated from a geometric viewpoint. Admitting an invariant manifold is a strong geometric property of a dynamical system. This text presents rigorous results on invariant manifolds and gives examples of possible applications. In the first part, discrete dynamical systems in Banach spaces are considered. Results on the existence and smoothness of attractive and repulsive invariant manifolds are derived. In addition, perturbations and approximations of the manifolds and the foliation of the adjacent space are treated. In the second part, analogous results for continuous dynamical systems in finite dimensions are established. In the third part, the theory developed is applied to problems in numerical analysis and to singularly perturbed systems of ordinary differential equations. The mathematical approach is based on the socalled graph transform, already used by Hadamard in 1901. The aim is to establish invariant manifold results in a simple setting that provides quantitative estimates. The book is targeted at researchers in the field of dynamical systems interested in precise theorems that are easy to apply. The application part might also serve as an underlying text for a student seminar in mathematics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Researchers interested in dynamical systems. Table of Contents Discrete Dynamical SystemsMaps  Existence
 Perturbation and approximation
 Smoothness
 Foliation
 Smoothness of the foliation with respect to the base point
Continuous Dynamical SystemsODEs  A general result for the timeT map
 Invariant manifold results
Applications  Fixed points and equilibria
 The onestep method associated to a linear multistep method
 Invariant manifolds for singularly perturbed ODEs
 RungeKutta methods applied to singularly perturbed ODEs
 Invariant curves of perturbed harmonic oscillators
 Blowup in singular perturbations
 Application of RungeKutta methods to differentialalgebraic equations
 Appendices
 Bibliography
 Index
