In this paper, we present a Bayesian framework for manoeuvring object tracking and intent prediction using novel α-stable Lévy state-space models, expressed in continuous time as Lévy processes. In contrast to conventional (fully) Gaussian formulations, the proposed models are driven by heavy-tailed α-stable noise and are thus much more able to capture extreme values/behaviours. This can better characterise sharp changes in the state, which may be induced by sudden and frequent manoeuvres such as swift turns or abrupt accelerations. In particular, they are represented in a conditionally Gaussian series form which ensures the tractability of the applied inference algorithms. A corresponding estimation strategy with the Rao-Blackwellised particle filter is then proposed and an efficient intent inference procedure is introduced. Here, the underlying intent, driving the target's long-term behaviour (e.g. reaching its final destination), is modelled as a latent variable. Real vessel data from maritime surveillance and human computer interactions (e.g. cursor data from motor-impaired interface users) are utilised to demonstrate the effectiveness of the proposed approach. It is shown to deliver noticeable improvements in the tracking and intent prediction performance (whenever relevant) compared with a more conventional Gaussian dynamic model.