Recent theoretical and empirical work exploits option-implied information to constrain the conditional moments of asset returns. In this paper, we characterize the information content that can be reliably used for this purpose under an option market's incompleteness. Our analysis relies on novel upper and lower bounds for a broad family of risk-neutral moments built from a given cross-section of option prices. We show the non-robustness of several risk-neutral moments in the literature, form new robust versions of them, and derive corresponding model-free bounds on the conditional moments of asset returns. These new bounds deviate substantially, both quantitatively and qualitatively, from those associated with non-robust risk-neutral moments.