Investor performance in financial markets can be significantly affected by their ability to model market volatility and correlation over time. The effectiveness of various market activities such as option pricing, portfolio optimisation and risk management rely on the accuracy of such modelling. This thesis proposes a series of multivariate GARCH models that attempt to accurately capture the volatility and correlation dynamics of stock returns. A Bayesian approach is utilised to estimate model parameters, extending classical maximum likelihood (ML) approaches commonly used in the literature for these types of models. A Bayesian prior distribution is proposed for a VECH model that expands the model parameter space and implicitly enforces necessary and sufficient conditions for its positive definiteness and covariance stationarity. An application to a set of US and UK stock indices supports this approach for both parameter and volatility estimation compared to classical ML applied to a competing BEKK model. Volatility asymmetry in stock returns is also discussed, and model selection techniques applied to an extended VECH model to determine the location and size of the asymmetry for international stock markets. In addition to asymmetry, an allowance is made for skewness and excess kurtosis in a proposed copula-GARCH model and is shown to exist in returns for an arbitrary stock portfolio. Moreover, the proposed model also performs well in estimating Value at Risk (VaR) for this portfolio, compared to other univariate and multivariate GARCH models considered. This thesis demonstrates the advantages of using the Bayesian approach for parameter estimation over classical ML, as well as the need to accurately capture the many properties of stock returns in order to improve the modelling of market volatility and correlation.