In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results: • Let ⋀ be a closed invariant set of f ϵ Diff¹(M). Then f|_⋀ is chain transitive and C¹-stably shadowing in a neighborhood of ⋀ if and only if ⋀ is a hyperbolic basic set (Theorem 4.2.1); • there is a residual set R ⊂ Diff¹(M) such that if f ϵ R and ⋀ is a locally maximal chain transitive set of f, then ⋀ is hyperbolic if and only if f |_⋀ is shadowing (Theorem 4.3.1).