  We initiate a systematic study of the perfection of affine group schemes of
finite type over fields of positive characteristic. The main result
intrinsically characterises and classifies the perfections of reductive groups,
and obtains a bijection with the set of classifying spaces of compact connected
Lie groups topologically localised away from the characteristic. We also study
the representations of perfectly reductive groups. We establish a highest
weight classification of simple modules, the decomposition into blocks, and
relate extension groups to those of the underlying abstract group.
