This thesis presents advances in theory and applications of mixture autoregressive (MAR) models in both Bayesian and frequentist frameworks. We improve the Bayesian analysis of mixture autoregressive models in the case of Gaussian components, by use of a sampling algorithm that allows to sample from the complete space of the posterior distribution of the parameters. In addition, we introduce a relabelling algorithm to deal with label switching, and propose density forecasts based on simulated Bayesian samples. We generalise the methodology to MAR models with Student-t mixture components, which includes Gaussian MAR as a limit case. We tackle the challenge of treating the number of degrees of freedom of the Student-t distribution as parameters whose posterior distribution has to be estimated. We propose using mixture vector autoregressive (MVAR) models for optimisation of portfolios of assets. The properties of MVAR models, combined with modern portfolio theory, allow in fact to analytically derive predictive distributions for portfolio returns at any time horizon. We also compare forecasting performance of MVAR models with other commonly used models in this context. We introduce an uncorrelated version of MAR models. By applying a set of linear constraints on the autoregressive parameters, the resulting model represents a direct alternative to GARCH models, as they both assume an uncorrelated but dependent structure for the data. We also propose an application of the uncorrelated MAR to residuals of an econometric model. All the data analysis is implemented in R, the majority of which is available in the package mixAR.