In this talk I will describe state-space models based on point process theory and Lévy processes, allowing very flexible modelling of continuous time non-Gaussian behaviours subject to irregular discrete time observations. In contrast with most of the classical models which use Brownian motion assumptions, our approach is based on pure jump-driven Lévy processes driving stochastic diferential equations, leading to powerful models based on, for example, alpha-stable or Generalised hyperbolic processes (including Student-t, variance-gamma and normal-inverse Gaussian). We are able to construct a full state-space model (The `Levy state-space model’) driven by such continuous time processes, observed at discrete time, as well as deriving central limit style theorems that prove Gaussianity of certain series residual terms, and inference for these models can be carried out using highly efficient Rao-Blackwellised versions of particle filters and sequential Markov chain Monte Carlo. The models can find application tracking of agile objects such as birds or drones, in financial prediction and in analysis of vibrational data under non-Gaussian perturbation.