In this paper we compare a set of different standard GARCH models with a group of Markov Regime-Switching GARCH (MRS-GARCH) in terms of their ability to forecast the US stock market volatility at horizons that range from one day to one month. To take into account the excessive persistence usually found in GARCH models that implies too smooth and too high volatility forecasts, in the MRS-GARCH models all parameters switch between a low and a high volatility regime. Both gaussian and fat-tailed conditional distributions for the residuals are assumed, and the degrees of freedom can also be state-dependent to capture possible time-varying kurtosis. The forecasting performances of the competing models are evaluated both with statistical and risk-management loss functions. Under statistical losses, we use both tests of equal predictive ability of the Diebold-Mariano-type and test of superior predictive ability. Under risk-management losses, we use a two-step selection procedure where we first check which models pass the tests of correct unconditional or conditional coverage and then we compare the best models under two subjective VaR-based loss functions. The empirical analysis demonstrates that MRS-GARCH models do really outperform all standard GARCH models in forecasting volatility at horizons shorter than one week under both statistical and VaR-based risk-management loss functions. In particular, all tests reject the presence of a better model than the MRS-GARCH with normal innovations. However, at forecast horizons longer than one week, standard asymmetric GARCH models tend to be superior.