There is a large body of literature investigating the statistical and economic significance of financial market predictability using return and volatility predictability. The results of these studies have been mixed and there is no consensus on the existence of predictability, particularly in the Australian context. Typically, these studies assume a mean-variance investor maximising their expected utility of wealth such that mean and variance are the only moments of returns considered in asset allocation. However, it has long been recognised that returns are not normally distributed. Using a Taylor series expansion to incorporate the higher moments of returns, this thesis examines whether the mean-variance criterion is appropriate for use in asset allocation under non-normality. This thesis demonstrates how optimal asset allocation can be implemented by considering higher moments. Both skewness and kurtosis are considered in the analysis, however the approach is easily extended to even higher moments if desired. The results suggest that where asset returns only moderately depart from normality, it remains appropriate to utilise the mean-variance framework for asset allocation. However, asset allocation utilising the approximated utility function with the third and fourth moments (skewness and kurtosis), may be superior to that of the mean-variance criterion in cases of large departures from normality. Various tests of robustness are conducted, yet this result that the use of higher moments in optimal asset allocation is, in some cases, superior to the mean-variance framework, remains.