In this chapter, we give either complete proofs or schemes of proof of the following main results: • If a diffeomorphism f of a smooth closed manifold has the Lipschitz shadowing property, then f is structurally stable (Theorem 2.3.1); • a diffeomorphism f has the Lipschitz periodic shadowing property if and only if f is Ω-stable (Theorem 2.4.1); • if a diffeomorphism f of class C² has the Hölder shadowing property on finite intervals with constants L;C; d₀; θ; ω, where θ ϵ (.1/2, 1) and θ+ω > 1, then f is structurally stable (Theorem 2.5.1); • there exists a homeomorphism of the interval that has the Lipschitz shadowing property and a nonisolated fixed point (Theorem 2.6.1); • if a vector field X has the Lipschitz shadowing property, then X is structurally stable (Theorem 2.7.1).