We review the shadowing theorem for compact hyperbolic sets of a diffeomorphism.Then we discuss the recent result of Pilyugin and Tikhomirov thatLipschitz shadowing implies hyperbolicity.We show how shadowing can beused to prove the existence of chaotic dynamics in the neighbourhood of atransversal homoclinic orbit.Next we show how how shadowing techniquescan be used to give computer-assisted proofs of the existence of transversalhomoclinic orbits.