Spatio-temporal models provide a mechanism for analysing data that occurs naturally in space and time such as pollution levels, functional magnetic resonance imaging data and temperature data. These models aim to capture the important features of the space time structure that can be overlooked by examining the spatial and temporal features separately. In this thesis a dynamic linear model (DLM) is used to describe a lattice Markov spatio-temporal system with Markov chain Monte Carlo (MCMC) techniques used to obtain estimates for the model parameters from the marginal posterior distributions. This thesis is concerned with the modelling of the latent structure of a Bayesian spatio-temporal model with a view to improving parameter inference, smoothing and prediction. The equilibrium distribution of a time stationary system will be examined, paying particular attention to edge effects and the effect of grid coarsening. In order to develop an effective MCMC algorithm the latent process is integrated out of the model. These techniques are illustrated using both simulated data and North Atlantic ocean temperature data.