Invariant manifolds and foliations have become very useful tools in dynamical systems. For infinite dimensional systems generated by evolutionary PDEs, the mere existence of these structures is non-trivial compared to those of ODEs due to issues such as the non-existence of backward (in time) solutions of some PDEs or nonlinear terms causing derivative losses. In addition to systematic generalization of the standard theory, often specific treatment has to be adopted based on the structure of the PDEs under consideration. We will briefly go through the general invariant manifold theory, followed by a few concrete PDEs. Also, applications to singular perturbations and homoclinic orbits for PDEs will be discussed. REFERENCES: 1. Dan Henry; Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes 840. 2. Chapters 1 and 6 at least. 1. A. Pazy; Semigroups of Linear Operators and Applications to PDEs, Springer Applied Math 44, 2. Chapters 1, 2, and 4. 3. R. Temam; Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Applied Math 68, Chapters 1-3. 4. P. Bates and C. Jones; Invariant Manifolds for Semilinear PDEs, Dynamics Reported V2, 1989. 5. K, Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Eqts, European Math Soc. 2011. 6. Unstable manifolds of Euler equations, Z. Lin and C. Zeng, 
avalaible at arxiv.org 7. Inviscid dynamical structures near Couette flow, Z. Lin and C. 
Zeng, ARMA and also avalaible at arxiv.org 8. Existence and persistence of invariant manifolds for semiflows in 
Banach space, P. Bates, K. Lu, and C. Zeng, Memoirs of AMS 9. Approximately invariant manifolds and global dynamics of spike 
states, P. Bates, K. Lu, and C. Zeng, Inventiones mathematicae 10. Invariant manifolds around soliton manifolds for the nonlinear ¨Klein-Gordon equation, Kenji Nakanishi, Wilhelm Schla, SIAM Math. Anal. and also avalaible at arxiv.org