In this thesis we consider some finite mixture time series models in which each component is following a well-known process, e.g. AR, ARMA or ARMA-GARCH process, with either normal-type errors or Student-t type errors. We develop MCMC methods and use them in the Bayesian analysis of these mixture models. We introduce some new models such as mixture of Student-t ARMA components and mixture of Student-t ARMA-GARCH components with complete Bayesian treatments. Moreover, we use component precision (instead of variance) with an additional hierarchical level which makes our model more consistent with the MCMC moves. We have implemented the proposed methods in R and give examples with real and simulated data.