This book presents an introduction to the geometric theory of infinite dimensional dynamical systems. Many of the fundamental results are presented for asymptotically smooth dynamical systems that have applications to functional differential equations as well as classes of dissipative partial differential equations. However, as in the earlier edition, the major emphasis is on retarded functional differential equations. This updated version also contains much material on neutral functional differential equations. The results in the earlier edition on Morse-Smale systems for maps are extended to a class of semiflows, which include retarded functional differential equations and parabolic partial differential equations.pFrom the reviews of the second edition:ppThis book presents a contemporary geometric theory of infinite-dimensional dynamical systems where the major emphasis is on retarded functional-differential equations. ??? Each chapter contains some abstract theorems but the authors give some examples as well illustrating these general results and having interesting applications. ??? This interesting book will be useful for researchers working in this field and, due to numerous examples, also for mathematicians working in applications. (Sergei A. Vakulenko, Mathematical Reviews, 2004 j)pThe first book, like the present one, is to a large extent devoted to functional differential equations. ??? The present editions of chapters that appeared in the first book, Invariant sets and attractors, Functional differential equations on manifolds, The dimension of the attractor, Attractor sets as C1-manifolds, The Kupka-Smale theorem, Conley index in noncompact spaces, are up-dated and contain additional examples. As the first book of the authors, the present one will be of interest and will be useful to a broad group of readers. (Peter Pol??cik, Zentralblatt MATH, Vol. 1002 (2), 2003)From the reviews of the second edition:pThis book presents a contemporary geo@K*=p£× ¾Û€