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Given integers g, m, and n, the heavy/light moduli space Mbar_{g, m|n} is a compactification of the moduli space of smooth (m+n)-marked curves of genus g. These spaces are particular examples of Hassett’s moduli spaces of weighted stable curves. Their rational cohomology gives a rich family of representations of products of symmetric groups. I’ll discuss recent work on the structure of this family of representations, and how they relate to the S_n-representations determined by the cohomology of Deligne-Mumford compactifications. This talk is based on joint work with Stefano Serpente and Claudia Yun.