Autoregressive Conditional Heteroskedastic (ARCH) and Generalised ARCH (GARCH) models are nonlinear volatility models designed to capture time-varying features of economic and financial time series. ARCH and GARCH models are usually estimated in a sampling theory framework using maximum likelihood techniques. However, there are a number of theoretical, implementation and analytical problems associated with this estimation procedure when applied to ARCH-type models. Fortunately, all of these problems can be addressed conveniently within a Bayesian framework. In the analysis of ARCH and GARCH models, Bayesian methods are preferred as they allow i) exact finite-sample results to be obtained for most ARCH and GARCH specifications, ii) inequality constraints to be easily imposed on parameters of a particular model, and iii) model uncertainty to be explicitly accounted for using model averaging. This dissertation proposes a full Bayesian analysis of ARCH and GARCH models, including parameter estimation, model averaging and volatility prediction. Implementation of the Bayesian approach requires the use of Markov chain Monte Carlo simulation methods. Illustrations are provided using data on the London Metal Exchange Index.