We evaluate how departure from normality may affect the allocation of assets. A Taylor series expansion of the expected utility allows to focus on certain moments and to compute the optimal portfolio allocation numerically. A decisive advantage of this approach is that it remains operational even for a large number of assets. While the mean-variance criterion provides a good approximation of the expected utility maximization under moderate non-normality, it may be ineffective under large departure from normality. In such cases, the three-moment or four-moment optimization strategies may provide a good approximation of the expected utility.