The computation of the cohomology of moduli spaces of n-pointed curves may essentially be reduced, by using the action of the symmetric group, to that of the cohomology of local systems on M_g for the symplectic group. The answer for genus 1 has been known for a long time. Van der Geer and I have obtained explicit formulas for genus 2, which are partly conjectural. Bergstr\"om has obtained results for genus 3. Consani and I have obtained a new construction of the cohomology in genus 1 related to cusp forms. I will discuss these results.