One of the main reasons for the high interest in hedge funds is their performance characteristics: Many hedge funds have yielded double-digit returns for their investors and, in some cases, in a fashion that seems uncorrelated with general market swings and with relatively low volatility. Several recent empirical studies, however, have challenged these characterizations of hedge fund returns, arguing that the standard methods of assessing their risks and rewards may be misleading. This monograph reviews the empirical facts surrounding hedge fund investments and proposes several new quantitative models for modeling hedge fund returns, risk exposures, and associated performance statistics.