This chapter deals mainly with the dimension theory of global attractors. We present some background and develop a relatively new approach which is based on some ideas due to O. Ladyzhenskaya (see Ladyzhenskaya [142] and the literature cited there) and assumes minimal smoothness properties of evolutions. We also discuss a wide class of dynamical systems which admits what is called the stabilizability (or quasi-stability) estimate. The notion of quasi-stability originally arose in the study of some plate models with nonlinear critical damping. However, the extension developed in this chapter allows us to consider a wider class of second order models (see Chapter 5) and also to cover several classes of parabolic and delayed models (see Chapters 4 and 6). In addition to attractors, other long-time behavior objects such as exponential attractors and determining functional sets are considered from the point of view of quasi-stability in this chapter.