A volatility model must be able to forecast volatility. This is the central requirement in almost all financial applications. There are two general classes of volatility models in widespread use. The first type formulates the conditional variance directly as a function of observables. The simplest examples are the autoregressive conditional heteroscedasticity (ARCH) and generalized autoregressive conditional heteroscedasticity (GARCH) models. The second general class formulates models of volatility that are not functions purely of observables. A number of stylized facts about the volatility of financial asset prices have emerged over the years. A good volatility model must be able to capture and reflect these stylized facts. To illustrate these stylized facts, data on the Dow Jones Industrial Index were used, and the ability of GARCH-type models was used to capture these features. Various aspects of the volatility process are important topics of research. The need for a model to forecast 100 or even 1000 steps into the future has suggested long memory or fractionally integrated processes. © 2007 Elsevier Ltd All rights reserved.