The response of a bistable dynamical system to a deterministic forcing is studied with emphasis on the kinetics of the passage across the barrier separating the two states, and compared to classical Kramers' theory describing the response to a Gaussian white noise forcing. The existence of nontrivial thresholds for the occurrence of transitions is established. Analytic results complemented by numerical simulations are derived for the characteristics of these transitions for periodic and chaotic forcings. The probabilistic properties of the response are finally addressed and some connections are established with the universal stable distributions of probability theory.