While the rational cohomology of arithmetic groups such as SL_n(Z) and Sp_{2n}(Z) can often be completely computed if the cohomological degree is small compared to n, little is known about it for high cohomological degrees. In this talk, I will discuss vanishing results that have recently been obtained for the high-dimensional rational cohomology of SL_n(Z), Sp_{2n}(Z) and other arithmetic Chevalley groups. This is related to a conjecture of Church--Farb--Putman and based on joint works with Brück--Miller--Patzt--Wilson, Brück--Patzt and Brück--Santos Rego.