The Gaussian assumption generally employed in many state-space models is usually not satisfied for real time series. Thus, in this work, a broad family of non-Gaussian models is defined by integrating and expanding previous work in the literature. The expansion is obtained at two levels: at the observational level, it allows for many distributions not previously considered, and at the latent state level, it involves an expanded specification for the system evolution. The class retains analytical availability of the marginal likelihood function, uncommon outside Gaussianity. This expansion considerably increases the applicability of the models and solves many previously existing problems such as long-term prediction, missing values and irregular temporal spacing. Inference about the state components can be performed because of the introduction of a new and exact smoothing procedure, in addition to filtered distributions. Inference for the hyperparameters is presented from the classical and Bayesian perspectives. The results seem to indicate competitive results of the models when compared with other non-Gaussian state-space models available. The methodology is applied to Gaussian and non-Gaussian dynamic linear models with time-varying means and variances and provides a computationally simple solution to inference in these models. The methodology is illustrated in a number of examples.