We use intraday data to construct measures of realized volatility, realized kurtosis, and realized skewness of returns of six major exchange rates vis-à-vis the dollar. The currencies under consideration are: (i) Australian dollar, (ii) Canadian dollar, (iii) Swiss franc, (iv) euro, (v) British pound, and (vi) Japanese yen. The period of the analysis spans from 1 July 2003 to 28 August 2015. We study in-sample and out-of-sample the predictive value of realized kurtosis and realized skewness for realized volatility, where we also differentiate between measures of upside realized volatility and downside realized volatility. We find that both realized kurtosis and realized skewness have in-sample predictive value in several models being studied. The out-of-sample results show that it is mainly realized kurtosis that helps to improve accuracy of one-day-ahead forecasts of realized volatility, but results depend on the assumed loss function and they differ across exchange rates.