Abstract semilinear functional differential equations arise from many biological, chemical, and physical systems which are characterized by both spatial and temporal variables and exhibit various spatio-temporal patterns.

1. Preliminaries.- 1.1 Semigroups and generators.- 1.2 Function spaces, elliptic operators, and maximal principles.- Bibliographical Notes.- 2. Existence and Compactness of Solution Semiflows.- 2.1 Existence and compactness.- 2.2 Local existence and global continuation.- 2.3 Extensions to neutral partial functional differential equations.- Bibliographical Notes.- 3. Generators and Decomposition of State Spaces for Linear Systems.- 3.1 Infinitesimal generators of solution semiflows of linear systems.- 3.2 Decomposition of state spaces by invariant subspaces.- 3.3 Computation of center, stable, and unstable subspaces.- 3.4 Extensions to equations with infinite delay.- 3.5 L2-stability and reduction of neutral equations.- Bibliographical Notes.- 4. Nonhomogeneous Systems and Linearized Stability.- 4.1 Dual operators and an alternative theorem.- 4.2 Variation of constants formula.- 4.3 Existence of periodic or almost periodic solutions.- 4.4 Principle of linearized stability.- 4.5 Fundamental transformations and representations of solutions.- Bibliographical Notes.- 5. Invariant Manifolds of Nonlinear Systems.- 5.1 Stable and unstable manifolds.- 5.2 Center manifolds.- 5.3 Flows on center manifolds.- 5.4 Global invariant manifolds of perturbed wave equations.- Bibliographical Notes.- 6. Hopf Bifurcations.- 6.1 Some classical Hopf bifurcation theorems for ODEs.- 6.2 Smooth local Hopf bifurcations: a special case.- 6.3 Some examples from population dynamics.- 6.4 Smooth local Hopf bifurcations: general situations.- 6.5 A topological global Hopf bifurcation theory.- 6.6 Global continuation of wave solutions.- Bibliographical Notes.- 7. Small and Large Diffusivity.- 7.1 Destablization of periodic solutions by small diffusivity.- 7.2 Large diffusivity under Neumann boundary conditions.- Bibliographical Notes.- 8. Invariance, Comparison, and Upper and Lower Solutions.- 8.1 Invariance and inequalities.- 8.2 Systems and strict inequalities.- 8.3 Applications to reaction diffusion equations with delay.- Bibliographical Notes.- 9. Convergence, Monotonicity, and Contracting Rectangles.- 9.1 Monotonicity and generic convergence.- 9.2 Stability and steady state solutions of quasimonotone systems.- 9.3 Comparison and convergence results.- 9.4 Applications to Lotka-Volterra competition models.- Bibliographical Notes.- 10. Dispativeness, Exponential Growth, and Invariance Principles.- 10.1 Point dispativeness in a scalar equation.- 10.2 Convergence in a scalar equation.- 10.3 Exponential growth in a scalar equation.- 10.4 An invariance principle.- Bibliographical Notes.- 11. Traveling Wave Solutions.- 11.1 Huxley nonlinearities and phase plane arguments.- 11.2 Delayed Fisher equation: sub-super solution method.- 11.3 Systems and monotone iteration method.- 11.4 Traveling oscillatory waves.- Bibliographical Notes.