This chapter evaluates the theoretical developments in ARCH modeling of time-varying conditional variances. It covers 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 model selection criteria. The chapter also discusses empirical regularities in asset returns, such as thick tails, volatility clustering, leverage effects, non-trading periods, forecastable events, volatility and serial correlation, co-movements in volatilities, and the relationship between macroeconomic variables and volatility. Additionally, it explores univariate parametric models like GARCH and EGARCH, and nonparametric and semiparametric methods for modeling asset returns. The chapter concludes with a discussion on the importance of capturing time-varying conditional variances in financial and economic models.This chapter evaluates the theoretical developments in ARCH modeling of time-varying conditional variances. It covers 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 model selection criteria. The chapter also discusses empirical regularities in asset returns, such as thick tails, volatility clustering, leverage effects, non-trading periods, forecastable events, volatility and serial correlation, co-movements in volatilities, and the relationship between macroeconomic variables and volatility. Additionally, it explores univariate parametric models like GARCH and EGARCH, and nonparametric and semiparametric methods for modeling asset returns. The chapter concludes with a discussion on the importance of capturing time-varying conditional variances in financial and economic models.