This paper develops a novel approach to modeling and forecasting realized volatility (RV) based on copula functions. Copula-based time series models can capture relevant characteristics of RV such as nonlinear dynamics and long-memory type behavior in a flexible yet parsimonious way. This makes it a possible contender to conventional approaches for modeling and forecasting realized volatility, such as the HAR-RV model of Corsi (2009). In an empirical application to daily realized volatility for the S&P500 index futures, we find that the copula-based model for RV (C-RV) outperforms the HAR-RV model for one-day ahead volatility forecasts in terms of accuracy and efficiency. Among the copula specifications considered, the Gumbel C-RV model achieves the best forecast performance, which highlights the importance of asymmetry and upper tail dependence for modeling volatility dynamics. Although we find substantial variation in the copula parameter estimates over time, conditional copulas improve the accuracy of volatility forecasts only marginally.