We study the robust measure of higher moments and employ their predictive power for future financial returns. In Chapter 1, we propose a new quantile-based measure of kurtosis. Then in Chapters 2 and 3, we provide an empirical analysis of the predictive information of robust volatility, robust skewness, and robust kurtosis for the cross-section of international stock index returns and cryptocurrency returns, respectively. In Chapter 1, we introduce robust kurtosis, which is a new quantile-based measure for the kurtosis of financial returns that is robust to outliers. Robust kurtosis is equivalent to the traditional moment-based kurtosis for the normal distribution, whereas for fat-tailed distributions it converges to the moment-based measure as the return horizon increases. We assess its asymptotic properties as well as its finite-sample properties under different distributional specifications. In Chapter 2, we provide an empirical analysis of the predictive information of robust conditional kurtosis for a large cross-section of international stock index returns. Using portfolio sorts and Fama-MacBeth cross-sectional regressions, we find that robust kurtosis carries a significant negative premium: higher robust kurtosis is related to lower future stock returns, especially for developed markets. This contrasts with the significant positive premium associated with robust skewness, especially for emerging markets. In Chapter 3, we provide an empirical analysis of the predictive information of robust conditional volatility, skewness, and kurtosis for a large cross-section of cryptocurrency returns. Using portfolio sorts and Fama-MacBeth cross-sectional regressions, we find that robust volatility carries a significant negative premium: higher robust volatility is related to lower future cryptocurrency returns, especially for one-week ahead returns. This contrasts with the significant positive premium associated with robust skewness, especially for one-day ahead returns.