This paper explores the structure of volatility on the JSE Securities Exchange of South Africa by employing ARCH-type models. Although the evidence suggests that volatility is prevalent on this market, it is established that the effects of shocks on volatility are symmetric, and that volatility is not a commonly priced factor. Hence, the standard GARCH(1,1) model provides the best description of the return dynamics relative to its complex augmentations. Further, application of the BDS test shows that the model significantly, but less than fully, accounts for non-linearities in the series.