Abstract
This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.