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We examine various phenomena induced by white Gaussian random perturbations in the response of non-linear dynamical systems. In the first part of this work digital and analog experiments are conducted on a simple single-degree-of-freedom oscillator with a piecewise linear restoring force and harmonic forcing. They reveal that small noise perturbations can give rise to large deviations of the response which ultimately lead to transitions between the coexisting attractors of the unperturbed system. These transitions are analyzed probabilistically by determining the mean time spent by the trajectories to exit from the basin of a given attractor. By determining the relationship between mean first-exit time and noise intensity, it is found that each attractor can be characterized by an activation energy which yields a measure of its relative stability. We also find that, even in the case of a single attractor, weak noise can induce large excursions to sets of the state space (chaotic semi-attractor) which are otherwise globally repelling in the absence of noise. In the second part of this work, some results obtained numerically are shown to be predicted theoretically by the use of asymptotic analyses of the randomly perturbed response of dynamical systems in the limit of weak noise. These techniques provide a generalization of the notion of potential to non-potential, non-equilibrium systems. In particular, the notion of activation energy is verified theoretically, and its determination may be possible without massive computer simulations.