This paper evaluates the out-of-sample forecasting accuracy of eleven models for weekly and monthly volatility in fourteen stock markets. Volatility is defined as within-week (within-month) standard deviation of continuously compounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997. The first half of the sample is retained for the estimation of parameters while the second half is for the forecast period. The following models are employed: a random walk model, a historical mean model, moving average models, weighted moving average models, exponentially weighted moving average models, an exponential smoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model and an EGARCH model. We first use the standard (symmetric) loss functions to evaluate the performance of the competing models: the mean error, the mean absolute error, the root mean squared error and the mean absolute percentage error. According to all of these standard loss functions, the exponential smoothing model provides superior forecasts of volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models. We also employ the asymmetric loss functions to penalise under/over-prediction. When under-predictions are penalised more heavily ARCH-type models provide the best forecasts while the random walk is worst. However, when over-predictions of volatility are penalised more heavily the exponential smoothing model performs best while the ARCH-type models are now universally found to be inferior forecasters.