  We compute the Hodge and de Rham cohomology of the classifying space BG
(defined as etale cohomology on the algebraic stack BG) for reductive groups G
over many fields, including fields of small characteristic. These calculations
have a direct relation with representation theory, yielding new results there.
The calculations are closely analogous to, but not always the same as, the
cohomology of classifying spaces in topology.
