This paper explores the impact of error-term non-normality on the performance of the normal-error Generalized Autoregressive Conditional Heteroskedastic (GARCH) model under small and moderate sample sizes. A non-normal-, asymmetric-error GARCH model is proposed, and its finite-sample performance is evaluated in comparison to the normal-error GARCH under various underlying error-term distributions. The results suggest that one must be skeptical of using the normal-error GARCH when there is evidence of conditional error-term non-normality. The conditional distribution of the error-term in a previous mainstream application of the normal GARCH is found to be non-normal and asymmetric. The same application is used to illustrate the advantages of the proposed non-normal-error GARCH model.