Abstract: The exact determination of the cohomology of SL_n(Z) with rational coefficients is a very difficult (and unsolved) problem for general n. I will discuss an approach to this problem that uses Borel-Serre duality, under which the cohomology in question is isomorphic to the homology of SL_n(Z) with coefficients in a dualizing module St, called the Steinberg module. The approach uses a resolution of St called the sharbly resolution. This enables the construction of hitherto unknown classes in the cohomology, and (joint with P. Patzt and J. Miller) a Hopf algebra structure on the direct sum (over n) of the cohomology groups.