This chapter evaluates the most important theoretical developments in ARCH type modeling of time-varying conditional variances. It covers the specification of univariate parametric ARCH models, general inference procedures, conditions for stationarity and ergodicity, continuous time methods, aggregation and forecasting of ARCH models, multivariate conditional covariance formulations, and the use of model selection criteria in an ARCH context. It also discusses the empirical regularities of financial market volatility, including thick tails, volatility clustering, leverage effects, non-trading periods, forecastable events, volatility and serial correlation, co-movements in volatilities, and macroeconomic variables and volatility. The chapter also presents a new conditional variance model for better characterizing stock return volatility, motivated by recent results on optimal filtering. The authors thank several individuals for their comments and acknowledge financial support from the National Science Foundation and the Center for Research in Security Prices. They also provide information on data inquiries and the availability of the GAIUSS™ code. The chapter includes a detailed discussion of the empirical regularities of asset returns, univariate parametric models, and inference procedures for ARCH models. It also discusses nonparametric and semiparametric methods for modeling volatility. The chapter concludes with a discussion of the implications of these findings for financial modeling and forecasting.This chapter evaluates the most important theoretical developments in ARCH type modeling of time-varying conditional variances. It covers the specification of univariate parametric ARCH models, general inference procedures, conditions for stationarity and ergodicity, continuous time methods, aggregation and forecasting of ARCH models, multivariate conditional covariance formulations, and the use of model selection criteria in an ARCH context. It also discusses the empirical regularities of financial market volatility, including thick tails, volatility clustering, leverage effects, non-trading periods, forecastable events, volatility and serial correlation, co-movements in volatilities, and macroeconomic variables and volatility. The chapter also presents a new conditional variance model for better characterizing stock return volatility, motivated by recent results on optimal filtering. The authors thank several individuals for their comments and acknowledge financial support from the National Science Foundation and the Center for Research in Security Prices. They also provide information on data inquiries and the availability of the GAIUSS™ code. The chapter includes a detailed discussion of the empirical regularities of asset returns, univariate parametric models, and inference procedures for ARCH models. It also discusses nonparametric and semiparametric methods for modeling volatility. The chapter concludes with a discussion of the implications of these findings for financial modeling and forecasting.