In this chapter, some basic concepts are introduced without proof. First, we introduce function spaces and infinite dimensional dynamical systems, which are helpful for understanding the discussion in Chap.  3 . Second, we introduce finite dimensional continuous-time dynamical systems defined by ODEs, which are used in Chap.  4 to study particle models. Finally, we introduce discrete-time dynamical systems defined by maps, which are used in Chap.  5 . See [27] for details of the materials in Sect. 2.1. If a reader is not yet initiated in functional analysis, [7] will prove useful, which is an introduction to functional analysis with applications to PDE. See [24, 43] for a description of dynamical systems and bifurcations, See [25, 82] for the center manifold theory.