  We survey old and new results about the cohomology of the moduli space $A_g$
of principally polarized abelian varieties of genus $g$ and its
compactifications. The main emphasis lies on the computation of the cohomology
for small genus and on stabilization results. We review both geometric and
representation theoretic approaches to the problem. The appendix provides a
detailed discussion of computational methods based on trace formulae and
automorphic representations, in particular Arthur's endoscopic classification
of automorphic representations for symplectic groups.
