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We present an outline of the theory of certain L\'evy-driven, multivariatestochastic processes, where the processes are represented by rational transferfunctions (Continuous-time AutoRegressive Moving Average or CARMA models) andtheir applications in non-Gaussian time series modelling. We discuss in detailtheir definition, their spectral representation, the equivalence to linearstate space models and further properties like the second order structure andthe tail behaviour under a heavy-tailed input. Furthermore, we study theestimation of the parameters using quasi-maximum likelihood estimates for theauto-regressive and moving average parameters, as well as how to estimate thedriving L\'evy process.

CARMA Processes driven by Non-Gaussian Noise