The relationship between short-term exposure to air pollution and mortality or morbidity has been the subject of much recent research, in which the standard method of analysis uses Poisson linear or additive models. In this paper, we use a Bayesian dynamic generalised linear model (DGLM) to estimate this relationship, which allows the standard linear or additive model to be extended in two ways: (i) the long-term trend and temporal correlation present in the health data can be modelled by an autoregressive process rather than a smooth function of calendar time; (ii) the effects of air pollution are allowed to evolve over time. The efficacy of these two extensions are investigated by applying a series of dynamic and non-dynamic models to air pollution and mortality data from Greater London. A Bayesian approach is taken throughout, and a Markov chain monte carlo simulation algorithm is presented for inference. An alternative likelihood based analysis is also presented, in order to allow a direct comparison with the only previous analysis of air pollution and health data using a DGLM.