State-space models have been successfully applied across a wide range of problems ranging from system control to target tracking and autonomous navigation. Their ubiquity stems from their modeling flexibility, as well as the development of a battery of powerful algorithms for estimating the state variables. For multivariate models, the Gaussian noise assumption is predominant due its convenient computational properties. In some cases, anyhow, this assumption breaks down and no longer holds. We propose a novel approach to extending the applicability of this class of models to a wider range of noise distributions without losing the computational advantages of the associated algorithms. The estimation methods we develop parallel the Kalman filter and thus are readily implemented and inherit the same order of complexity. We derive all of the equations and algorithms from first principles. In order to validate the performance of our approach, we present specific instances of non-Gaussian state-space models and test their performance on experiments with synthetic and real data.