Rationality properties of complex characters of finite groups

What can one say about the fields of values of irreducible complex characters χχ of a given finite group GG? In particular when GG is a finite (quasi)simple group? What about the ``McKay situation'', i.e. when the degree of
χχ is coprime to a fixed prime pp? We will discuss these and related questions, and report on some recent results, most of which are obtained jointly with G. Navarro.