Group cohomology of arithmetic groups is ubiquitous in the study of arithmetic K-theory and algebraic number theory. Rationally, SL_n(Z) and its finite index subgroups don't have cohomology above dimension n choose 2. Using Borel-Serre duality, one has access to the high dimensions. Church, Farb, and Putman conjectured that the high dimensional cohomology of SL_n(Z) with trivial rational coefficients vanishes. In this lecture series, we will give an introduction to these notions, prove the aforementioned conjecture in codimensions 0 and 1. We will also study the top cohomology of principal congruence subgroups. In the final lecture, we summarize some further directions and open problems in the field.