For any prime $p$, a finite group has as many irreducible complex characters of degree prime to $p$ as the normalizer of a Sylow $p$-subgroup. This equality, conjectured by John McKay in 1971, was reduced in 2007 by Isaacs--Malle--Navarro to a conjecture on representations of finite simple groups. Thanks to their classification we know that the latter are essentially finite groups of Lie type. Deligne--Lusztig theory helps to prove the McKay conjecture by this approach. For groups of characteristic different from $p$, the normalizers of Sylow $p$-subgroups belong to a larger class of subgroups related to parabolic subgroups of the ambient algebraic group for which Deligne-Lusztig varieties and induction functors have been used in the 1990s to provide a substitute to parabolic induction. This works well for unipotent characters. An important step is the construction of a Jordan decomposition of characters which is equivariant with respect to automorphisms of the simple group.