In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distribution that give nature too much freedom to in ict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], we prove that certain worst-case expectation constraints are computationally tractable under these new ambiguity sets. We showcase the practical applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.