Since the early 1970s, mathematicians have tried to extend the work of N. Fenichel and of M. Hirsch, C. Pugh and M. Shub to give conditions under which invariant manifolds for semiflows persist under perturbation of the semiflow. This work provides natural conditions and establishes the desired theorem. The technique is geometric in nature, and in addition to rigorous proofs, an informal outline of the approach is given with useful illustrations. This book features: important theoretical tools for working with infinite-dimensional dynamical systems, such as PDEs; previously unpublished results; and new ideas regarding invariant manifolds.