  We prove the McKay conjecture on characters of odd degree. A major step in
the proof is the verification of the inductive McKay condition for groups of
Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal
normal abelian subgroup is contained in a maximally split torus by means of a
new equivariant version of Harish-Chandra induction. Specifics of characters of
odd degree, namely that they only lie in very particular Harish-Chandra series
then allow us to deduce from it the McKay conjecture for the prime~$2$, hence
for characters of odd degree.
