This paper explores Mixed Data Sampling (MIDAS) regression models, which allow for time series data sampled at different frequencies. The focus is on volatility and related processes, but the method has broader applications in macroeconomics and finance. The paper introduces various lag structures to parameterize regressions parsimoniously and relates them to existing models. It also proposes new extensions of the MIDAS framework. An empirical section provides further evidence on the risk-return tradeoff and microstructure noise in volatility forecasting. The MIDAS regression model combines recent developments in volatility estimation with distributed lag models. It allows left-hand and right-hand side variables to be sampled at different frequencies. The paper discusses several polynomial specifications, including exponential Almon and beta lags, and their implications for parameter estimation and model flexibility. It also addresses infinite polynomials and autoregressive augmentations, as well as stepfunction-based MIDAS regressions. The paper highlights the advantages of MIDAS models in capturing rich dynamics of high-frequency processes in a simple and parsimonious manner. It discusses various extensions, including nonlinear MIDAS regressions, tick-by-tick applications, and multivariate MIDAS models. These models are shown to be flexible and useful for addressing complex issues in volatility forecasting and risk-return tradeoff analysis. The paper revisits two empirical applications: the risk-return tradeoff and volatility prediction. It finds a robustly positive and statistically significant risk-return tradeoff across horizons and predictors. It also provides empirical evidence on the impact of microstructure noise on volatility prediction, showing that corrections for noise can improve forecasting accuracy. The paper concludes that MIDAS regressions are a powerful tool for analyzing mixed-frequency data, offering flexibility, parsimony, and the ability to capture complex dynamics in financial and economic data. The results suggest that MIDAS models can be effectively used to estimate risk-return relationships and forecast volatility, even in the presence of microstructure noise.This paper explores Mixed Data Sampling (MIDAS) regression models, which allow for time series data sampled at different frequencies. The focus is on volatility and related processes, but the method has broader applications in macroeconomics and finance. The paper introduces various lag structures to parameterize regressions parsimoniously and relates them to existing models. It also proposes new extensions of the MIDAS framework. An empirical section provides further evidence on the risk-return tradeoff and microstructure noise in volatility forecasting. The MIDAS regression model combines recent developments in volatility estimation with distributed lag models. It allows left-hand and right-hand side variables to be sampled at different frequencies. The paper discusses several polynomial specifications, including exponential Almon and beta lags, and their implications for parameter estimation and model flexibility. It also addresses infinite polynomials and autoregressive augmentations, as well as stepfunction-based MIDAS regressions. The paper highlights the advantages of MIDAS models in capturing rich dynamics of high-frequency processes in a simple and parsimonious manner. It discusses various extensions, including nonlinear MIDAS regressions, tick-by-tick applications, and multivariate MIDAS models. These models are shown to be flexible and useful for addressing complex issues in volatility forecasting and risk-return tradeoff analysis. The paper revisits two empirical applications: the risk-return tradeoff and volatility prediction. It finds a robustly positive and statistically significant risk-return tradeoff across horizons and predictors. It also provides empirical evidence on the impact of microstructure noise on volatility prediction, showing that corrections for noise can improve forecasting accuracy. The paper concludes that MIDAS regressions are a powerful tool for analyzing mixed-frequency data, offering flexibility, parsimony, and the ability to capture complex dynamics in financial and economic data. The results suggest that MIDAS models can be effectively used to estimate risk-return relationships and forecast volatility, even in the presence of microstructure noise.