In this preliminary chapter, we define pseudotrajectories and various shadowing properties for dynamical systems with discrete and continuous time (Sects. 1.1 and 1.2), study the notion of chain transitivity (Sect. 1.1), describe hyperbolicity, Ω-stability, and structural stability (Sect. 1.3), and prove a lemma on finite Lipschitz shadowing in a neighborhood of a hyperbolic set (Sect. 1.4).