Heterostructures of thin-film perovskite ferroelectrics and relaxors were grown by in situ pulsed laser deposition on La 0.5 Sr 0.5 CoO 3 /MgO and Pt/sapphire. The dielectric response of the heterostructures was measured as a function of frequency, temperature, amplitude of ac field, magnitude of dc field, and varying the measurement rate. In the ferroelectric state, the existence of ferroelectric domains and the fast change of their configuration were found to deternine the dynamic dielectric nonlinearity and the dc dielectric hysteresis. In thin-film relaxors, the dynamic dielectric nonlinearity was shown to be related to the orientation of the randomly interacting dipoles in a random field. No indications of the domain-type dynamics were detected even at low temperatures and/or under an applied dc field. A possible relaxational mechanism responsible for the dielectric hysteresis in thin-film relaxors is suggested and discussed. In thin-film (Ba,Sr)TiO 3 , a relaxational high-temperature dielectric hysteresis was observed.