In this paper we document that realized variation measures constructed from high-frequency returns reveal a large degree of volatility risk in stock and index returns, where we characterize volatility risk by the extent to which forecasting errors in realized volatility are substantive. Even though returns standardized by ex post quadratic variation measures are nearly gaussian, this unpredictability brings considerably more uncertainty to the empirically relevant ex ante distribution of returns. Carefully modeling this volatility risk is fundamental. We propose a dually asymmetric realized volatility (DARV) model, which incorporates the important fact that realized volatility series are systematically more volatile in high volatility periods. Returns in this framework display time varying volatility, skewness and kurtosis. We provide a detailed account of the empirical advantages of the model using data on the S&P 500 index and eight other indexes and stocks.