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The aim of this survey paper is to provide an account of some of the important developments in the autoregressive conditional heteroskedasticity (ARCH) model since its inception in a seminal paper by Engle (1982). This model takes account of many observed properties of asset prices, and therefore, various interpretations can be attributed to it. We start with the basic ARCH models and discuss their different interpretations. ARCH models have been generalized in different directions to accommodate more and more features of the real world. We provide a comprehensive treatment of many of the extensions of the original ARCH model. Next we discuss estimation and testing for ARCH models and note that these models lead to some interesting and unique problems. There have been numerous applications and we mention some of these as we present different models. The paper includes a glossary of the acronyms for the models we describe.

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