Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk (CVaR) constraints. This talk discusses risk constraints when the distributional information of the uncertain parameters consists of moment information (e.g., mean, covariance, support) and certain structural information, for which we mention two specific examples: logconcavity and dominance on the tail. We find that the ambiguous risk constraints in these settings can be recast or approximated using conic constraints that facilitate computation. Finally, we demonstrate the theoretical results via case studies on power system operation and appointment scheduling.