The purpose of this proposal is to further the study of the cohomology of the moduli space of curves. Moduli spaces play a fundamental role in modern algebraic geometry and among them, the moduli space of curves distinguishes itself for several reasons. It is also of great importance in complex geometry, differential geometry, and theoretical physics (via string theory). Specifically, we want to study the cohomology of the moduli spaces of smooth or stable n-pointed curves of genus 3. We will take the natural action of the symmetric group into account. Working with Euler characteristics of compactly supported cohomology, we may as well consider those of symplectic local systems on M_3; the local systems come from the moduli space A_3 of principally polarized abelian threefolds and correspond to irreducible representations of the symplectic group Sp(6,Z) indexed by a descending triple (a,b,c) of nonnegative integers. In case a+b+c is even, the answer for A_3 determines that for M_3; and we have a conjectural answer for which there is much evidence. An important goal is to find a formula for the case a+b+c odd and to prove it; A_3 is no longer of any help. In this way, the situation resembles that in genus > 3, which we also want to begin to explore. We have convincing guesses for all (a,b,c) with sum < 20, with only 3 exceptions. Of particular interest are the `motives associated to SO(7) and SO(9) that we have found and the corresponding vector-valued Teichmüller modular forms.