In this talk, we discuss several recent results on relations between various shadowing properties of dynamical systems with discrete and continuous time and structural stability. Main topics: (1) C1-interiors of sets of dynamical systems with various shadowing properties; (2) shadowing properties of smooth vector fields and structural stability; (3) equivalence of Lipschitz shadowing and structural stability; (4) relations between periodic shadowing and Omega-stability. The formal seminar will be followed by an informal more technical discussion for anyone who is interested in the details of the work.